North Carolina Math 1

CustomStudent Workbook

Unit 2: Linear Functions

North CarolinaMath 1

This book is licensed for a single student’s use only. The reproduction of any part, for any purpose, is strictly prohibited.

© Common Core State Standards. Copyright 2010. National Governor’s Association Center for Best Practices and

Council of Chief State School Offi cers. All rights reserved.

1 2 3 4 5 6 7 8 9 10

ISBN 978-0-8251-9007-0

Copyright © 2020

J. Weston Walch, Publisher

Portland, ME 04103

www.walch.com

Printed in the United States of America

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Table of Contents

iii

Unit 2: Linear FunctionsLesson 2.2: Interpreting Linear Expressions (A–SSE.1b★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-1Lesson 2.3: Connecting Graphs and Equations of Linear Functions (F–IF.6★) . . . . . . . . . . . . . U2-11Lesson 2.4: Finding the Slope or Rate of Change of Linear Functions (F–IF.6★) . . . . . . . . . . . U2-24Lesson 2.5: Calculate and Interpret the Average Rate of Change (F–IF.6★) . . . . . . . . . . . . . . . . U2-34Lesson 2.6: Interpreting Parameters (F–LE.5★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-46Lesson 2.7: Graphing the Set of All Solutions (A–REI.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-54Lesson 2.8: Graphing Linear Equations in Two Variables (A–CED.2★) . . . . . . . . . . . . . . . . . . . U2-67Lesson 2.9: Solving Linear Inequalities in Two Variables (A–REI.12) . . . . . . . . . . . . . . . . . . . . . U2-77Lesson 2.10: Key Features of Linear Functions (F–IF.4★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-87Lesson 2.11: Graphing Linear Functions (F–IF.7★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-97Lesson 2.12: Comparing Linear Functions (F–IF.9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-110Lesson 2.13: Building Functions from Context (F–BF.1a★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-130Lesson 2.14: Arithmetic Sequences (F–BF.2★) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U2-141

Station ActivitiesSet 1: Comparing Linear Models (A–CED.2★, A–REI.10, F–IF.7★) . . . . . . . . . . . . . . . . . . . . . . U2-149Set 2: Relations Versus Functions/Domain and Range (F–BF.1a★, F–IF.1, F–IF.2) . . . . . . . . . U2-156

Coordinate Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CP-1Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F-1Bilingual Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G-1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Martie

Sticky Note

Marked set by Martie

Martie

Sticky Note


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v

The North Carolina Math 1 Custom Student Workbook includes all of the student pages from the Teacher Resource necessary for your day-to-day classroom use. This includes:

• Warm-Ups

• Problem-Based Tasks

• Practice Problems

• Station Activity Worksheets

In addition, it provides Guided Practice examples that parallel the examples in the Teacher Resource. This supports:

• Taking notes during class

• Working problems for preview or additional practice

The workbook includes the first Guided Practice example with step-by-step prompts for solving, and the remaining Guided Practice examples without prompts.

At the end of the workbook, you will find a reference section with blank coordinate planes for graphing, useful formulas, and a bilingual glossary.

The workbook is printed on perforated paper so you can submit your assignments and three-hole punched to let you store it in a binder.

Introduction

U2-1

Date:Name:UNIT 2 • LINEAR FUNCTIONS A–SSE.1b★

Lesson 2.2: Interpreting Linear Expressions

© Walch Education© Walch Education© Walch Education© Walch Education North Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1North Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1 Custom Student WorkbookCustom Student WorkbookNorth Carolina Math 1 Custom Student Workbook2.2

Warm-Up 2.2Javier deposited $750 in a bank account that earns interest at a rate of 3% of his initial deposit each year. He left the money in the account for 5 years. Use this information to complete the problems that follow. Explain your answers. Use the following formula for simple interest: I = prt, where I represents I represents Ithe interest earned, p represents the principal (the money Javier invested), r represents the interest r represents the interest rrate, and t represents the time in years.t represents the time in years.t

1. How much interest did Javier earn in 5 years?

2. How much money was in Javier’s account after 5 years?


U2-2

UNIT 2 • LINEAR FUNCTIONS A–SSE.1b★


Name: Date:

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Use the given information to solve each problem.

1. Is 5(2 + x) equal to 10 + 2x? Why or why not?

2. Is 5(3 + x) equal to 15 + 5x? Why or why not?

3. Is (3 + 2)x equal to 5x? Why or why not?

4. In the expression 10 + 2x, what effect does the variable x have on the independent term 10?

5. What happens as x becomes larger in the expression 3x?

continued

Scaffolded Practice 2.2: Interpreting Linear Expressions

U2-3




6. What happens as x becomes smaller, but not 0, in the expression x

3?

7. In the expression 5x

y, what effect does increasing the value of x have on the expression?

8. In the expression x(5 + y), what effect does increasing the value of x have on y, if any?

9. For what values of x will the result of 5x be greater than 25?

10. For what values of x will the result of 5x be smaller than 5?

U2-4



Name: Date:


Guided Practice 2.2Guided Practice 2.2Example 1

A new car loses an average value of $1,800 per year. When Nia bought her new car, she paid $25,000. The expression 25,000 – 1800yThe expression 25,000 – 1800yThe expression 25,000 – 1800 represents the current value of the car, where y represents the number of years since Nia bought it. What effect, if any, does the change in the number of years since Nia bought the car have on the original price of the car?

1. Refer to the expression given: 25,000 – 1800yRefer to the expression given: 25,000 – 1800yRefer to the expression given: 25,000 – 1800 .

2. Determine the effect that the number of years has on the original price of the car.

continued

U2-5




Example 2

To calculate the perimeter of an isosceles triangle, the expression 2s + b is used, where s represents the length of the two congruent sides and b represents the length of the base. What effect, if any, does increasing the length of the congruent sides have on the expression?

Example 3

Money deposited in a bank account earns interest on the initial amount deposited as well as any interest earned as time passes. This simple interest can be described by the expression P(1 + rn), where P represents the initial amount deposited, P represents the initial amount deposited, P r represents the interest rate, and r represents the interest rate, and r n represents the number of years that pass. How does a change in each variable affect the value of the expression?

U2-6



Name: Date:


Problem-Based Task 2.2: Searching for a Greater SavingsProblem-Based Task 2.2: Searching for a Greater SavingsAustin plans to open a savings account. The amount of money in a savings account can be found by using the equation s = p(1 + rt), where rt), where rt p is the principal, or the original amount deposited into the account; r is the rate of interest; and r is the rate of interest; and r t is the amount of time. t is the amount of time. tAustin is considering two savings accounts. He will deposit $1,000 as the principal into either account. In Account A, the interest rate will be 0.015 per year for a term of 5 years. In Account B, the interest rate will be 0.02 per year for a term of 3 years. Which account has more money at the end of its term? If he could, assuming the interest rates stay the same, would it be wise for Austin to leave his money in the account that has less savings for an additional year? Explain your reasoning.less savings for an additional year? Explain your reasoning.

If he could, assuming the

interest rates stay the same, would it be wise for Austin to leave his money

in the account that has less savings for an additional year?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

U2-7




Use your understanding of terms, coefficients, factors, exponents, and the order of operations to answer each of the following questions.

1. Is the expression x5 3

2+

always equal to the expression 4x? Explain your answer.

2. Is the expression 2(3 + x) equal to the expression 6 + 3x? Explain your answer.

3. Is the expression (5 • 2) x equal to the expression 10 x ? Explain your answer.

4. A transfer station charges $15 for a waste disposal permit and an additional $5 for each cubic yard of garbage it disposes of. This relationship can be described using the expression 15 + 5x. What effect, if any, does changing the value of x have on the cost of the permit?

5. Absolute Cable company bills on a monthly basis. Each bill includes a $30.00 service fee plus $4.75 in taxes and $2.99 for each movie purchased. The following expression describes the cost of the cable service per month: 34.75 + 2.99m. If Absolute Cable lowers the service fee, how will the expression change?

continued

Practice 2.2: Interpreting Linear Expressions AA

U2-8



Name: Date:


6. In order for a pet to lose weight in a healthy manner, a veterinarian suggested an overweight large-breed dog lose 2 pounds per week. If the expression x – 2y – 2y – 2 represents this situation, what must be true about the value of y?

7. The product of 7, x, and y is represented by the expression 7xy. If the value of x is negative, what can be said about the value of y in order for the product to remain positive?

8. A bank account balance for an account with an initial deposit of P dollars earns interest at P dollars earns interest at Pan annual rate of r. The amount of money in the account after n years is described using the following expression: P(1 + rn). What effect, if any, does decreasing the value of r have on the r have on the ramount of money after n years?

9. For what values of x will the result of –3(–3x – 4) be greater than 3?

10. A tire can hold C cubic feet of air. It loses a set amount of its air during each period of time, C cubic feet of air. It loses a set amount of its air during each period of time, C t. This rate of loss, written as a decimal, is r. This situation can be described using the following formula: formula: CC(1 – (1 – C(1 – CC(1 – C rtrt). What effect, if any, does increasing the value of ). What effect, if any, does increasing the value of rt). What effect, if any, does increasing the value of rtrt). What effect, if any, does increasing the value of rt rr have on the value of have on the value of r have on the value of rr have on the value of r CC ? ?C ?CC ?C

U2-9




Practice 2.2: Interpreting Linear Expressions BUse your understanding of terms, coefficients, factors, exponents, and the order of operations to complete each of the following problems.

1. Explain why the expression 7 • 3 x is not equal to the expression 21 x.

2. Explain why the expression (5 • 2) x is equal to the expression 10 x.

3. Julio and his sister bought 8 books and m magazines for $1 each, and then they split the cost.

The amount of money that Julio spent is represented by the expression m1

2(8 )+ . Does the

number of books purchased affect the value of m?

4. Satellite Cell Phone company bills on a monthly basis. Each bill includes a $19.95 service fee for 500 minutes plus a $3.95 communication tax and $0.15 for each minute over 500 minutes. The following expression describes the cost of the cellphone service per month: 23.90 + 0.15m. If Satellite Cell Phone lowers its service fee, how will the expression change?

5. The expression x

9 is given. Describe the value of this expression if the value of x is less than 1,

but greater than 0.

continued

U2-10



Name: Date:


6. For what values of x will the result of 0.5(10 + x) be greater than 7?

7. A bank account balance for an account with an initial deposit of P dollars earns interest at P dollars earns interest at Pan annual rate of r. The amount of money in the account after n years is described using the following expression: P(1 + rn). What effect, if any, does increasing the value of r have on the r have on the ramount of money after n years?

8. The effectiveness of an initial dose, d, of a particular medicine decreases over a period of time, t, at a certain percentage rate, r, written as a decimal. This situation can be described using the expression: d(1 – rt). What effect, if any, does decreasing the value of rt). What effect, if any, does decreasing the value of rt r have on the value of r have on the value of r d ? d ? d

9. The population of a town changes at a rate of r each year. To determine the number of r each year. To determine the number of rpeople after n years, the following expression is used: P(1 + rn), where P represents the initial P represents the initial Ppopulation, r represents the rate, and n represents the number of years. If the population were declining, what values would you expect for the factor (1 + rn)?

10. Explain why the expression 3y Explain why the expression 3y Explain why the expression 3 (3x + 5) is equal to the expression 9xy + 15y + 15y + 15 .

U2-11© Walch Education© Walch Education North Carolina Math 1North Carolina Math 1 Custom Student WorkbookCustom Student Workbook

2.3

Date:Name:UNIT 2 • LINEAR FUNCTIONS F–IF.6★

Lesson 2.3: Connecting Graphs and Equations of Linear Functions

Warm-Up 2.3The following graph shows the approximate United States population from 1900 to 2010, as recorded by the U.S. Census Bu reau.

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100

75

5019001890 1910 1920 1930 1940

Census year

U.S. Population by Census

1950 1960 1970 1980 1990 2000 2010 2020

Popu

latio

n (in

mill

ions

)

1. What was the rate of change in the population from 1900 to 2000? Is this greater or less than the rate of change in the population from 2000 to 2010?

2. Which 10-year time periods have the highest and the lowest rates of change? How did you find these?

3. What do you predict the U.S. population will be in 2020? Explain your reasoning.


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2.32.3

UNIT 2 • LINEAR FUNCTIONS F–IF.6★


Name: Date:

Use a graphing calculator to graph each of the odd-numbered problems. Then answer each of the questions that follow.

1. Graph the equation 5x + 2y + 2y + 2 = 0.

x

y

2. What are the slope and y-intercept of the equation?

3. Graph the equation 2x – 4 = y.

x

y


continued

Scaffolded Practice 2.3: Connecting Graphs and Equations of Linear Functions


2.3



continued

5. Graph the equation –xGraph the equation –xGraph the equation – + 1 = y.

x

y


7. Graph the equation 6x – y = 6.

x

y



2.32.3



Name: Date:


x

y



2.3



Guided Practice 2.3Examp le 1

The graph of a linear function is shown. Use two points on the line and the formula 2 1

2 1

my y

x x=

−−

to find

its slope. Write the equation for this line using the point-slope form, y – y1 = m(x(x( – x – x x1). Then, rewrite the

result in slope-intercept form, y = mx + mx + mx b. Use th e slope-intercept form to determine the y-intercept, b.

– 10 – 8 – 6 – 4 – 2 2 4 6 8 10

10

8

6

4

2

– 2

– 4

– 6

– 8

– 10

y

x0

1. Determine the slope of the line.

2. Use the point-slope form to write the equation for this line.

3. Rewrite the equation in slope-intercept form.

4. Determine the y-intercept of the line.

continued


2.32.3



Name: Date:

Example 2

Use the following graph to determine t he slope and the y-intercept of the line.

– 5 – 4 – 3 – 2 – 1 1 2 3 4 5

x

5

4

3

2

1

– 1

– 2

– 3

– 4

– 5

y

(2.5, –0.5)

(1, 0.25)

0


2.3



Problem-Based T ask 2.3: The Boiling Point of WaterThe boiling point of water is a function of the altitude. This means that water’s boiling point changes depending on its distance above or below sea level. The table shows several altitudes in feet above sea level and the boiling point of water at each altitude. Find the linear equation that models this situation, and create a graph of the function. Using your graph, what is the estimated boiling point of water at an altitude of 20,000 feet above sea level?

Altitude (ft) Boiling point (°F)2,000 208.13,500 205.36,500 199.6

Using your graph, what is the

estimated boiling point of water

at an altitude of 20,000 feet above

sea level?

SMP1 ✓ 2 ✓3 4 ✓5 ✓ 6 7 ✓ 8


2.32.3



Name: Date:

Practic e 2.3: Connecting Graphs and Equations of Linear FunctionsUse what you know about linear functions to complete problems 1–8.

1. Write the equation in slope-intercept form for the linear function in the graph.

20

4 6 8 10

8

6

4

2

–2

–1

y

x

2. The graph of a linear function has a slope of 3 and contains the point (15, 6). What is the y-intercept?

3. Which of the following linear functions has the greater y-intercept: the line containing the points (30, 40) and (40, 60), or the line containing the points (30, 40) and (40, 61)? Explain.

4. The mass of a package of 50 mints, including the container, is 131 grams. If half of the mints are removed, the total mass is 81 grams. If x is the mass of one mint and y is the total mass, what linear function describes the total mass?

continued

AA


2.3



5. The graph of a linear function is a horizontal line. When x = –3, y = 4. What is the equation for this linear function? Explain.

6. Two points on the graph of a linear function are (–4, 5) and (–6, 9). What is the slope-intercept form of the equation of the line?

7. The perimeter of a frame with a given width is a linear function of the height of the frame. Several frames have the same width. One has a height of 60 cm and a perimeter of 240 cm. Another has a height of 90 cm and a perimeter of 300 cm. What is the perimeter as a linear function of the height? What does the y-intercept represent?

8. What is the slope-intercept equation of a line through the points (–8, –4) and (–2, –11)?

Use the following information for problems 9 and 10.

Olives are sold at the supermarket salad bar. A customer scoops the olives into a container and pays by weight, but the price he pays is reduced to account for the weight of the container (the tare weight). The prices for two different total weights are given in the table.

Weight (oz) Price ($)4 2.009 5.50

9. What linear function can be used to find the price for x ounces of olives?

10. What does the y-intercept represent in terms of the given scenario?


2.32.3



Name: Date:

Practice 2.3: Connecting Graphs and Equations of Linear FunctionsThe following graph shows the amount of paint needed to paint the doors of a house. Use the graph to answer questions 1 and 2.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Am

ount

of p

aint

(gal

lons

)

Number of doors

1. What is the approximate rate of change for the interval [2, 7]?


B

continued


2.3



The following graph shows the value of the U.S. dollar compared to the value of the Australian dollar on a specific day. Use the graph to answer questions 3–5.

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

5

10

15

20

25

30

35

40

45

50

55

60

Valu

e of

Aus

tral

ian

dolla

r

Value of U.S. dollar



5. Could you predict the rate of change for a third interval on the same graph? If so, what is your prediction?

continued


2.32.3



Name: Date:

Each year, volunteers at a three-day music festival record the number of people who camp on the festival grounds. The following graph shows the number of campers for each of the last 20 years. Use the graph to answer questions 6 and 7.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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1000

1500

2000

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3000

3500

4000

Year

Tota

l num

ber o

f cam

pers



continued


2.3



The following graph shows the yearly population of a small town. Use the graph to answer questions 8–10.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Years

Popu

latio

n



10. How does the rate of change differ for each interval?

U2-24

Name: Date:

North Carolina Math 1North Carolina Math 1North Carolina Math 1North Carolina Math 1 Custom Student WorkbookCustom Student Workbook Custom Student WorkbookCustom Student Workbook Custom Student Workbook2.4

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Lesson 2.4: Finding the Slope or Rate of Change of Linear Functions

Warm-Up 2.4Lupita wants to buy a boat that will have the best resale value after 3 years.

1. At one boat dealer, she found a boat she likes that sells for $15,000 and depreciates at a rate of 30% per year. What will be the value of the boat after 3 years?

2. At another dealer, she found a boat that costs $12,000 and depreciates at a rate of 20% per year. What will be the value of the boat after 3 years?

3. Which boat will have the greater value in 3 years?


Martie

Sticky Note


U2-25

Date:Name:

North Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1North Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1 Custom Student WorkbookNorth Carolina Math 1 Custom Student WorkbookCustom Student WorkbookNorth Carolina Math 1 Custom Student Workbook2.4




Find the slope between the two given points.

1. (2, 7) and (1, 3)−

2. ( 2, 1)− and (5, 4)

3. (8, 0) and (0, 8)

4. (0, 6) and ( 4, 4)− −

5. (2, 3) and (2, 3)−

continued

Scaffolded Practice 2.4: Finding the Slope or Rate of Change of Linear Functions

U2-26

Name: Date:





6. (10, 1) and ( 3, 1)−

7. (7, 5) and (6, 15)

8. (3.7, 4) and (1.7, 9)

9. ( 12, 6)− and (4, 6)

10. (9, 4) and (14, 7)

U2-27

Date:Name:






Marc gets paid $15 per lawn he mows. Graph the proportional relationship. Determine the slope and what it means in the context of the problem. How can the slope be used to determine how many lawns Marc mowed if he made $180? What is the equation that describes the relationship between the two quantities?

1. Create a table to show how the two quantities described vary.

2. Graph the proportional relat ionship.

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50

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125

75

25

225

y

x0

350

300

325

275

2 4 6 8 101 3 5 7 9 11 12 13 14 15 16 17 18 19 20

3. Determine the sl ope and what it means in the context of the problem.

4. How can the slope be used to determine how many lawns Marc mowed if he made $180?

5. Write the equation that describes the relationship between the two quantities.

continued

U2-28

Name: Date:





Example 2

At a roadside farm stand, you can buy 5 pounds of any of the vegetables for a total cost of $6. Determine the slope of the line formed by the proportional relationship between the number of pounds purchased and the cost of the vegetables. Explain what the slope means in the context of the problem. Finally, use the slope to determine how many pounds of vegetables can be purchased for $13. Assume there is no sales tax.

Example 3

A new plumber has just started his own business. In order to try and gain customers, he is running a special for his services. He charges $16 per hour, plus a standard house call fee of $25. Determine the slope of the line that passes through the points of the total cost for jobs lasting from 2 hours to 6 hours. Explain what the slope means in the context of the problem. Finally, use the slope to determine how many hours of work a customer could get for $150. Assume there is no sales tax.

U2-29

Date:Name:





Problem-Based Task 2.4: Pondering the PondProblem-Based Task 2.4: Pondering the PondFelix has purchased a piece of property that has a 3,500-gallon pond. He wants to drain the pond and fill it in so he can build a garage. Felix has hired a pump truck to remove the water from his pond. The pump truck crew members take turns monitoring the level of water over the course of several hours. The following graph shows the number of gallons of water per hour being pumped out of the pond.

2 4 6 8 10

200

1 3 5 7 9

y

x0

2,600

2,400

2,200

2,000

1,800

1,600

1,400

1,200

1,000

400

600

800

11 12 13 14 15 16 17 18 19 20

2,800

3,600

3,400

3,200

3,000

Tim

e (h

ours

)

Number of gallons

Determine the hourly rate at which the pond is being drained. Use this rate to write the equation of the line in the graph. Assuming the pump truck’s hose is draining constantly at the same rate, how long will it take to empty the entire pond? There are 8 hours in 1 workday. If Felix can only afford to pay the pump truck crew for 2 workdays, will he be able to empty his pond? Explain your reasoning.pay the pump truck crew for 2 workdays, will he be able to empty his pond? Explain your reasoning.

If Felix can only afford to pay the pump truck crew for 2 workdays, will he be able to empty his pond?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

U2-30

Name: Date:





Practice 2.4: Fi nding the Slope or Rate of Change of Linear Functions Practice 2.4: Fi nding the Slope or Rate of Change of Linear Functions For problems 1–4, determine the unit rate of the two quantities described and write what that rate means in the context of the problem. Then write the equation that describes the relationship between the two quantities.

1. A family pack of 12 tacos costs $8.

2. A baseball player throws the ball 180 feet in 3 seconds.

3. Charonika types 900 words in 12 minutes.

4. There are 640 chairs in 16 rows.

For problems 5–10, graph the relationship between the given quantities, and then use the slope of the line to answer the question.

5. A financial adviser meets with 8 clients each day. How many clients would she meet in 12 days?

6. Christy drove 1,088 miles on a road trip in 16 hours. How many miles did she drive per hour? Assume that she drove a constant rate of speed for the entire trip.

7. Mario rode his bike 103.6 miles in 7 days. If he rode the same number of miles daily, how many miles did he ride per day?

8. A fishing store gives customers 10 new lures for every 2 fishing poles they buy. How many lures would the store give to a customer who bought 8 fishing poles?

9. Alexander ordered 6 large pepperoni pizzas. The total cost was $82.50, which included a delivery fee of $7.50. How much did each pizza cost, not including the delivery fee?

10. Sheila started her savings account with $28, and saved the same amount of money each month. After 6 months, she had $220 in the account. How much did Sheila save each month, not including her starting amount?

AA

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Practice 2.4: Finding the Slope or Rate of Change of Linear FunctionsFor problems 1–5, calculate the rate of change for each scenario described.

1. The fuel capacity of a popular hybrid car is 11.9 gallons. The function for this situation is f(f(f x(x( ) = –0.02x + 11.9, where x represents miles and f(f(f x(x( ) represents the amount of fuel remaining. What is the rate of change for this scenario?

2. The cost of videotaping a basketball tournament is modeled by the function f(f(f x(x( ) = 20x + 350, where x represents the cost of each video. What is the rate of change for this scenario?

3. An investment of $750 is invested at a rate of 3.5%, compounded monthly. The function that

models this situation is f xx

= +

( ) 750 1

0.035

12

12

, where x represents time in years. What is the

rate of change for the interval [2, 7]?

4. The price of a stock started out at $23 and has declined to 25% of its value every 2 weeks. The

function that models this decline is f xx

=( ) 150(0.25) 2 , where x represents time in weeks. What

is the rate of change for the interval [3, 6]?

5. The conversion of inches to centimeters follows a function. Several conversions are listed in the table. What is the rate of change for this function?

Inches (x)x)x Centimeters (fCentimeters (fCentimeters ( (f(f x))x))x5 12.7

10 25.415 38.120 50.825 63.5

B

continued

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The following table represents the total cost to ship a package based on the package’s weight in pounds. Use the table to answer questions 6 and 7.

Number of pounds (x)x)x Total cost in dollars (fTotal cost in dollars (fTotal cost in dollars ( (f(f x))x))x0 5.255 5.90

10 6.5515 7.2020 7.85

6. What is the rate of change for this function over the interval [0, 10]?


Use the given information to complete problem 8.

8. A Petri dish starts out with 9 bacteria. The number of bacteria doubles every 3 minutes. Use the table to calculate the rate of change for the interval [3, 12].

Minutes (x)x)x Number of bacteria (fNumber of bacteria (fNumber of bacteria ( (f(f x))x))x0 93 186 369 72

12 144

continued

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The following table represents the worth each year of an initial investment of $650 that earns 3.4% interest compounded quarterly. Use the table to answer questions 9 and 10.

Years (x)x)x Investment value in dollars (fInvestment value in dollars (fInvestment value in dollars ( (f(f x))x))x0 6502 695.544 744.276 796.418 852.20



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Lesson 2.5: Calculate and Interpret the Average Rate of Change

Name: Date:

North Carolina Math 1North Carolina Math 1North Carolina Math 1North Carolina Math 1 Custom Student WorkbookCustom Student Workbook Custom Student WorkbookCustom Student Workbook Custom Student Workbook2.52.52.52.5


Warm-Up 2.5Bus drivers practice their routes before the first day of school to make sure every student will arrive home in a timely manner. This graph shows a bus’s distance over time on a practice run, where the point (60, 30) represents your bus stop.

20 40 60 80

40

30

20

10

35

25

15

5

0

45

10 30 50 70 90

Distance (miles)

Time (minutes)

1. Describe the route the bus took in relation to miles over time (in minutes).

2. Use the points (0, 0) and (60, 30) to find the average speed of the bus in miles per hour.

3. Will the bus get you home in a timely manner? Explain what factors may cause the average speed of the bus to increase or decrease.


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Use the given information to complete problems 1–5.

1. A jet ski has a fuel capacity of 17 gallons. The function that represents how the amount of fuel changes as a function of distance ridden is f(f(f x(x( ) = –1.5x + 17, where x represents miles ridden and f(f(f x(x( ) represents the amount of fuel remaini ng. What is the rate of change for this scenario?

2. The cost of hiring a wedding photographer is modeled by the function f(f(f x(x( ) = 40x + 200, where x represents the number of hours worked and f(x represents the number of hours worked and f(x represents the number of hours worked and f( ) represents the final cost. What is the rate of change for this scenario?

3. An investment of $825 is invested at a rate of 2.9%, compounded monthly. The function that

models this situation is ( ) 825 10.029

12

12

f xx

= +

, where x represents time in years, and f(f(f x(x( )

represents investment value. What is the rate of change for the interval 4 ≤ x ≤ 8?

4. The price of a stock started at $75 per share and has declined to 50% of its value every 3 weeks.

The function that models this decline is ( ) 75 0.50 3f xx

( )= , where x represents time in weeks.

What is the rate of change for the interval 2 ≤ x ≤ 7?

continued

Scaffolded Practice 2.5: Calculate and Interpret the Average Rate of Change

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5. The conversion of pounds to kilograms follows a function. Several conversions are listed in the table. What is the rate of change for this function?

Pounds (x)x)x Kilograms (fKilograms (fKilograms ( (f(f x))x))x

5 2.25

10 4.5

15 6.75

20 9

25 11.25

The following table represents the total cost to rent a bouncy house for a birthday party. Use the table to complete problems 6 and 7.

Number of hours (x)x)x Total cost in dollars (fTotal cost in dollars (fTotal cost in dollars ( (f(f x))x))x

0 48

1 80

2 112

3 144

4 176

6. What is the rate of change for this function over the interval 0 ≤ x ≤ 2?

7. What is the rate of change for this function over the interval 2 ≤ x ≤ 4?

continued

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Use the given information to complete problem 8.

8. A Petri dish starts out with 17 bacteria. The number of bacteria doubles every 8 minutes. Use the table to calculate the rate of change for the interval 8 ≤ x ≤ 24.

Minutes (x)x)x Number of bacteria (fNumber of bacteria (fNumber of bacteria ( (f(f x))x))x

0 17

8 34

16 68

24 136

32 272

The following table represents the worth each year of an initial investment of $325 that earns 1.7% annual interest, compounded quarterly. Use the table to complete problems 9 and 10.

Year (x)x)x Amount (fAmount (fAmount ( (f(f x))x))x

0 325

3 341.97

6 359.82

9 378.61

12 398.37

9. What is the average rate of change for this function over the interval 0 ≤ x ≤ 9?

10. What is the average rate of change for this function over the interval 6 ≤ x ≤ 12?

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Guided Practice 2.5Ex ample 1

Gilligan captains a submarine that is at a depth of 150 feet below sea level. He dives the submarine to a depth that is 12 times the original depth in 45 seconds. Describe the submarine’s rate of change during this 45-second interval.

1. Determine the new depth of the submarine.

2. Calculate the rate of change.

continued

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Ex ample 2

The following gra ph shows the distance in miles, m, that Theresa hiked from her campsite in h hours. Identify the hourly interval with the largest rate of change, and explain what it means in the context of the problem.

1 2 3 4 5 6

4

3

2

1

0

5

6

Hours (h)

Mile

s (m

)

Exa mple 3

The following table illustrates the average amount of time a person spends sleeping each night as he or she ages. What is the average rate of change, in minutes per year, that a person spends sleeping each night from age 10 to age 70?

Age (years)

Average night’s sleep (minutes)

10 57020 54030 51040 48050 45060 42070 390

continued

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Example 4

Find the average rate of change over the interval (–1, 0). What does the average rate of change tell you about the function on the interval? Does the rate of change for the function appear to increase, decrease, or remain the same as x increases greater than 0?

1 2 3 4 5

–1

–2

–3

–4

–5

4

3

2

1

0

5y

x

–1–2–3–4–5

Example 5

Find the average rate of change for each of the following functions over the given interval. Then, write a conclusion regarding how you can use the average rate of change to compare the three functions on the given interval.

• f (f (f x(x( ) = 2x + 4 from x = 2 to x = 3

• g(x(x( ) = x2 + 6 from x = 2 to x = 3

• h(x(x( ) = 4x from x = 2 to x = 3

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The polar ice caps are melting, and ecologists have found that the polar bear population is suffering. The current population of polar bears worldwide is about 26,000, and is declining by about 20% each decade. This is represented by the function f (f (f x(x( ) = 26,000(0.8)x. Calculate the rates of change for the polar bear population for two domains: the next 5 decades, and decades 5 through 10. In which domain is the polar bear population decreasing more quickly? Why is that the case?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

Prob lem-Based Task 2.5: Polar Bear Population Decline

In which domain is the polar bear population decreasing more quickly? Why is that the case?

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Use the interval ≤ ≤x2 5 to find the average rate of change in problems 1–3.

1. = −f x x( ) 2 3

2. = + −f x x x( ) 4 12

3. =f x x( ) 2(3 )

4. Find the average rate of change in the following table on the interval of ≤ ≤x0 3 .

x 0 1 2 3 4f (f (f x)x)x 3 6 12 24 48

5. Use the function =f x x( ) 3 to determine which of the following intervals has the greatest average rate of change: ≤ ≤x0 1 , ≤ ≤x1 2 , or ≤ ≤x2 3 . Predict what will happen when the interval is ≤ ≤x9 10 .

The following table lists the high temperatures (T ) in Charlotte, N.C., for the first 10 days (D) in Charlotte, N.C., for the first 10 days (D) in Charlotte, N.C., for the first 10 days ( ) of February 2017. Use the table to complete problems 6 and 7.

D 1 2 3 4 5 6 7 8 9 10T 73 67 54 45 62 68 73 66 61 53

6. Find the average rate of change in temperature for all 10 days.

7. Which interval has the fastest decrease in temperature? Which interval had the fastest increase in temperature?

Practice 2.5: Calculate and Interpret the Average Rate of Change A

continued

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Use the following information and graph to complete problems 8–10.

A ball tossed in the air from ground level is modeled by the function = −h t t t( ) 144 16 2 , where h is the height in feet of the ball in the air and t is the time in seconds.

2 4 6 8 10

300

200

100

250

150

50

01 3 5 7 9

y

x

–1

(4.5, 324)

8. On what time interval will the ball’s height in the air decrease?

9. Find the average rate of change from the launch to the ball’s maximum height in the air.

10. Compare the average rate of change on the intervals ≤ ≤x0 4.5 and ≤ ≤x4.5 9 . Do you expect the rate of change to be the same for both intervals? Explain your reasoning.

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Use the interval ≤ ≤x1 4 to find the average rate of change in problems 1–3.

1. = −f x x( ) 3 4

2. = + −f x x x( ) 3 42

3. =f x x( ) 2(2 )

4. Find the average rate of change in the following table on the interval of ≤ ≤x0 3 .

x 0 1 2 3 4f (f (f x)x)x 2 4 8 16 32

5. Use =f x x( ) 4 to determine which of the following intervals has the greatest average rate of change: ≤ ≤x0 1 , ≤ ≤x1 2 , or ≤ ≤x2 3 . Predict what will happen when the interval is

≤ ≤x9 10 .

The following table lists the hourly temperatures (t) in Charlotte, N.C., for the number of hours (t) in Charlotte, N.C., for the number of hours (t h) since midnight on Feb. 22, 2017. Use the table to complete problems 6 and 7.

h 0 1 2 3 4 5 6 7 8 9 10t 57 57 56 54 54 53 52 53 55 59 64

6. Find the average rate of change in temperature from 12 A.M. to 10 A.M.

7. Which interval had the fastest decrease in temperature? Which interval has the fastest increase?

Practice 2.5: Calculate and Interpret the Average Rate of Change B

continued

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Use the following information and graph to complete problems 8–10.

The height of a toy rocket launched from the ground can be modeled by the function =− +h t t t( ) 16 642 , where h is the height in feet of the rocket in the air and t is the

time in seconds.

1 2 3 4 5

60

70

40

20

50

30

10

0

y

x

(2, 64)

8. On what time interval will the rocket’s height in the air decrease?

9. Find the average rate of change from the launch to the rocket’s maximum height in the air.

10. Compare the average rate of change on the intervals ≤ ≤x0 2 and ≤ ≤x2 4 . Do you expect the rate of change to be the same for both intervals? Explain your reasoning.

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UNIT 2 • LINEAR FUNCTIONS F–LE.5★

Lesson 2.6: Interpreting Parameters

Name: Date:



Warm-Up 2.6You are buying a membership to a gaming store. The membership fee is $5 per month and each game costs $2 to rent. There are no late fees.

1. Write a linear function to represent the amount of mon ey you spend in a month.

2. W hat is the domain of this function?

3. Graph the function and identify the y-intercept.

4. What does the y-intercept represent in the context of the problem?

5. What does the slope represent in the context of the problem?


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Date:Name:UNIT 2 • LINEAR FUNCTIONS F–LE.5★




Identify the parameters of the functions in problems 1–5.

1. f(f(f x(x( ) = 2x + 3

2. f(f(f x(x( ) = 3x + 2

3. f(f(f x(x( ) = –3x + 10

4. f(f(f x(x( ) = 4x – 2

5. f(f(f x(x( ) = –4x – 8

Use what you know about functions to complete problems 6–10.

6. You join a spa. Each massage costs $15 and each pedicure costs $20. What equation represents the total amount spent at the spa? What are the parameters in this scenario?

continued

Scaffolded Practice 2.6: Interpreting Parameters

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7. You join a cycle class. The monthly membership fee is $20 plus the rate per hour spent cycling is $4. What function represents this scenario? What are the parameters in this scenario?

8. Tom subscribed to a movie rental program. He pays a monthly fee of $8.00, plus $1.50 for each movie rented. What are the parameters in this scenario?

9. The number of pigs on a farm in a video game is described by f (f (f x(x( ) = 2x + 24, where x is time in

hours. What do the numbers 2 and 24 tell you about the number of pigs on the virtual farm?

10. Emily hides $300 in her mattress and deposits $400 into an account every year. What is the function that represents the amount of money that Emily has? What are the parameters?

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continued

Guided Practice 2.6Example 1

You visit a pick-your-own apple orchard. There is an entrance fee of $5.00, plus you pay $0.50 for each apple you pick. Write a function to represent this scenario. Complete a table of values to show your total cost if you pick 10, 20, 30, 40, and 50 apples. Graph the function and identify the parameters in this problem. What do the parameters represent in the context of the problem?

1. Write a function.

2. Create a table.

3. Identify the domain of the function and then graph the function.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

5

10

15

20

25

30

35

Number of apples picked

Tota

l cos

t ($)

4. Identify the parameters.

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Example 2

Sam is mowing lawns to make extra money to buy a car. For every mowing job, he charges an initial fee of $10 plus $12 for each hour of work. His total fee for an average yard that takes 2 hours to cut is $34. Write a function to show how much he charges for mowing lawns that take 30 minutes, 1 hour, 3 hours, and 5 hours. Use the function to create a table, then graph the function. In context of the problem, interpret the following parameters: slope and y-intercept.

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Problem-Based Task 2.6: Cell Phone PlansProblem-Based Task 2.6: Cell Phone PlansY ou are comparing cell phone providers in order to determine which one offers the best deal. AT&Me offers a plan with a monthly fee of $25, plus a $0.10 per minute charge. Tracmyphone offers a plan with a monthly fee of $15, with a $0.15 per minute charge. You typically use fewer than 100 minutes per month. Which plan would be the best choice for you and why?choice for you and why?

Which plan would be the best choice for you and why?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

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Practice 2.6: Interpreting ParametersIdentify the parameters for the functions in problems 1–5.

1. f(f(f x(x( ) = 7x + 5

2. f(f(f x(x( ) = 2x + 3

3. f(f(f x(x( ) = –2x + 10

4. f(f(f x(x( ) = 2(3 + x)

5. f(f(f x(x( ) = 3(2 + x) + 5


6. You join a gym. The monthly membership fee is $10 and the rate per hour is $2. What is the function that represents this scenario? What are the parameters in this scenario?

7. Claire subscribes to a movie rental program. She pays a monthly fee of $5.00, plus $1.25 for each movie rented. What are the parameters in this scenario?

8. Max is picking apples with his brother. The number of apples in his bag is described by f(f(f x(x( ) = 18x + 15, where x is the number of minutes Max spends picking apples. What do the numbers 18 and 15 tell you about Max’s apple picking?

9. The number of ants in an ant farm is described by f(f(f x(x( ) = 10x + 2, where x is time in hours. What do the numbers 10 and 2 tell you about the number of ants in the colony?

10. Anastasia hides $200 in her mattress and deposits $150 into an account every year. What is the function that represents the amount of money that Anastasia has? What are the parameters?

AA

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Practice 2.6: Interpreting ParametersIdentify the parameters for the functions in problems 1–5.

1. f(f(f x(x( ) = 3x + 12

2. f(f(f x(x( ) = 4x – 8

3. f(f(f x(x( ) = –6x + 13

4. f(f(f x(x( ) = 5(2 + x)

5. f(f(f x(x( ) = 2(4 + x) + 9


6. Your aunt hides $100 in her mattress and deposits $300 into an account every year. What is the function that represents this scenario? What are the parameters?

7. Lily subscribes to a game rental program. She pays a monthly fee of $7.00 plus $2.50 for each game rented. What are the parameters in this scenario?

8. You join a gym. The monthly membership fee is $12.00 and the rate per hour of gym use is $3.75. What are the parameters in this scenario?

9. Kendall is picking strawberries with his sister. The number of strawberries in his basket is described by f(f(f x(x( ) = 35x + 20, where x is the number of minutes Kendall spends picking strawberries. What do the numbers 35 and 20 tell you about Kendall’s strawberry picking?

10. You have an ant farm. The number of ants in your colony is described by f(f(f x(x( ) = 25 + 3x, where x is in days. What do the numbers 25 and 3 tell you about the number of ants in your colony?

AAAB

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UNIT 2 • LINEAR FUNCTIONS A–REI.10Lesson 2.7: Graphing the Set of All Solutions

Warm-Up 2.7Mallory had $1 this morning and asked her mom for another dollar. Instead of giving Mallory more money, her mom said she would give Mallory $2 the next day if she still had the dollar she was holding. Plus, she would continue to give Mallory $2 each day as long as she saved it all. Mallory agreed to the deal and wondered how much money she might have at the end of the week. To find out, Mallory graphed the equation y = 2x + 1, as shown, where x represents the number of days and y represents the amount of money she wou ld have.

2 4 6 8 10

20

1819

16

14

12

10

8

9

6

4

2

0

x

y

Days

Dol

lars

1 3 5 7 9

17

15

13

11

7

5

3

1

1. If Day 0 is Monday, how much could Mallory have on Wednesday?

2. How much could Mallory have on Friday?

3. Explain how the equation and graph represent Mallory receiving $2 each day.

Lesson 2.7: Graphing the Set of All Solutions

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Graph each of the odd-numbered problems. Then name two points that lie on the line and satisfy the equation in each of the even-numbered problems.

1. Graph the equation 5x + 2y + 2y + 2 = 0.

x

y

2. What are two points that lie on the line and satisfy the equation?


x

y


continued

Scaffolded Practice 2.7: Graphing the Set of All Solutions

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continued

5. Graph the equation –xGraph the equation –xGraph the equation – + 1 = y.

x

y


7. Graph the equation 6x – y = 6.

x

y


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x

y


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Graph the solution set for the linear equation –3x + y = –2.

1. Solve the equation for y.

2. Make a table. Choose at least three values for x and find the corresponding values of y using the given equation.

3. Plot the ordered pairs on the coordinate plane.

2 4 6 8 10

10

1 3 5 7 9

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

4. Connect the points by drawing a line through them. Use arrows at each end of the line to show that the line continues indefinitely in each direction. This represents all of the solutions for the equation.equation.

continued

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Example 2

Graph the solution set for the equation y = 3x.

Example 3

The Russell family is driving 1,000 miles to the beach for their summer vacation. Mr. Russell drives at an average rate of 60 miles per hour and plans on stopping four times to break up the journey. Let t represent t represent tthe number of hours the Russells will travel before they reach their destination, and let d represent the d represent the dremaining distance after each stop. Write an equation in terms of d and d and d t that represents the 1,000-mile t that represents the 1,000-mile ttrip. Next, draw a graph that represents the number of miles traveled for each hour of the trip.trip. Next, draw a graph that represents the number of miles traveled for each hour of the trip.

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Problem-Based Task 2.7: Saving for Colle geJake graduated from high school and is working at the family furniture store to save for college. He can either be paid $12.50 per hour or earn a commission of 15% on all his sales above $500. Jake’s commission is represented by the equation c = (0.15)(c = (0.15)(c s – 500), where c is Jake’s commission in dollars and c is Jake’s commission in dollars and c s is the amount of Jake’s total sales. Create a second equation describing Jake’s hourly wages, w, in terms of the number of hours he works, h. Graph each equation and describe Jake’s earning potential based on the two types of wages.

Graph each equation and

describe Jake’s equation and

describe Jake’s equation and

earning potential based on the two types of wages.

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

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continued

Practice 2.7: Graphing the Set of All SolutionsFor problems 1–4, draw the graph that represents the solution set of the equation.

1. 2x + y = –1

2. 4x – 2y – 2y – 2 = –6

3. y = 2x

4. y = 3x

For problems 5 and 6, use each given graph to find three solutions that will satisfy the equation.

5. yx1

2=

–10 –8 –6 –4 –2 2 4 6 8 10

14

12

10

8

6

4

2

– 2

0

– 4

– 1

– 3

– 5

–1–9 –7 –5 –3 1 3 5 7 9

1

9

7

5

3

15

13

11

y

x

AA

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continued

6. y x1

87=− +

–10 –8 –6 –4 –2 2 4 6 8 10

14

12

10

8

6

4

2

– 2

0

– 4

– 1

– 3

– 5

–1–9 –7 –5 –3 1 3 5 7 9

1

9

7

5

3

15

13

11

y

x

For problems 7–10, use the given information to answer the questions.

7. A company’s yearly profit during its first 5 years of operation can be modeled by the equation P = 225(1.13)P = 225(1.13)P x + 400, where x is the number of years since the company started and P is the P is the Pprofit in dollars. Draw a graph to represent this situation. If this pattern continues, what would the company’s profit be in year 7?

8. Katya is a caterer. She has a cookie recipe that calls for 2 eggs per batch. Katya wants to know the number of eggs she needs according to how many batches she cooks. What equation can be used to represent the number of eggs Katya needs for any number of batches? Draw a graph to represent this situation. How many eggs would Katya need for 4 batches of cookies?

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9. The price of a certain company’s stock grew at the same rate during the first 6 months of the year. The following table shows the price in dollars per share of stock, P, for each mo nth, m.

m P1 35.752 363 36.254 36.55 36.756 37

What equation can be used to represent this situation? Draw a graph to represent the growth of the stock’s price per share during this period. If the pattern continues, what will the price per share be after 12 months?

10. Gas costs $3 per gallon, which can be modeled by the equation C = 3C = 3C x, where x is the number of gallons needed and C is the total cost in dollars. Your car has a 15-gallon gas tank. Draw a graph C is the total cost in dollars. Your car has a 15-gallon gas tank. Draw a graph Cthat represents how much you might have to pay depending on the number of gallons of gas you buy. How much will you have to pay for 7.5 gallons of gas?

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Practice 2.7: Graphing the Set of All SolutionsFor problems 1–4, draw the graph that represents the solution set of the equation.

1. 3x + 2y + 2y + 2 = 2

2. x – y = 4

3. y = 3x

4. yx1

4=

For problems 5 and 6, use the given graph to find three solutions that will satisfy the equation.

5. –3x + 2y + 2y + 2 = 4

2 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10 4

continued

B

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6. yx1

3=

2 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10 4

For problems 7–10, use the given information to answer the questions.

7. A house painter starts a job with 65 gallons of paint, and uses 5 gallons every hour. Draw the graph that represents all solutions for this situation. If he started 6 hours ago, how many gallons of paint should he have left?

8. Certain bacteria in a science lab grow at the rate of yx

(mass in grams) • 215= , where x is in hours. If there were 0.1 gram of bacteria to start with, how many grams of bacteria were there 60 hours later? Draw the graph that represents all solutions for this situation.

continued

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Name: Date:




9. Enrico wants to bike twice as far he did the previous day for 5 days straight. Draw the graph that represents the number of miles Enrico bikes each day. If Enrico biked 3 miles the first da y, how many miles must he go on the fifth day? Assume that the first day is day 0.

10. Mr. Samuelson spent $3,000 on a new, more efficient air conditioning unit for his large house. He hopes to save 35% each month on his electricity bill, which averages $350 a month. The equation that represents this situation is y = 3000 – (0.35)(350)x, where x represents the number of months and y represents the difference betw een what Mr. Samuelson spent on the air conditioning unit and the money he hopes to save each month, in dollars. How many months will it take for his total savings to be greater than the $3,000 he spent? Draw the graph that represents all solutions for this situation.

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Date:Name:UNIT 2 • LINEAR FUNCTIONS A–CED.2★

Lesson 2.8: Graphing Linear Equations in Two Variables



Warm-Up 2.8Read the information that follows and use it to complete the problems.

A cell phone company charges a $20 flat fee plus $0.05 for every minute used for calls.

1. Make a table of values from 0 to 60 minutes in 10-minute intervals that represents the total amount charged.

2. Write an algebraic equation that could be used to represent the situation.

3. What do the unknown values in your equation represent?


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UNIT 2 • LINEAR FUNCTIONS A–CED.2★


Name: Date:



Use the given information to solve each problem.

1. Rick charges $50 plus $40 an hour to fix cars. Let x be the number of hours that he works and ybe the overall cost that he charges. Write the equation.

2. Graph the equation from the previous problem.

3. Rachel is a veterinarian. She charges $20 as a base and $10 for each animal that she vaccinates. Let x be the number of animals vaccinated and x be the number of animals vaccinated and x y be the total amount charged. Write the equation.


Scaffolded Practice 2.8: Graphing Linear Equations in Two Variables

continued

U2-69





5. Pilar has a side job as a dog walker. She charges a flat rate of $20 per walk, plus $5 for every dog she takes. Let x be the number of dogs walked and y be the total cost. Write the equation.


7. Jake tutors students. He charges $10 for the first session and $5 for each additional session. Let xbe the number of sessions after the first, and y be the total cost. Write the equation for cost y.


continued

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9. Rylie cleans carpets. She charges $30 for the job and $15 per hour. Let x be the number of hours spent on the job and y be the total cost. Write the equation.


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continued


A local convenience store owner spent $10 on 144 pencils to resell at the store. What is the equation for the store’s profit if each pencil sells for $0.50? Graph the equation using a table of values.

1. Read the problem and then reread the problem, determining the known quantities.

2. Identify the slope and the y-intercept.

3. Substitute the slope and y-intercept into the equation y = mx + b, where m is the slope and b is the y-intercept.

4. Change the slope into a fraction in preparation for graphing.

5. Rewrite the equation using the fraction.

6. Set up the coordinate plane and identify the independent and dependent variables.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

15

-15

-14

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

11

12

13

14

y

x

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7. Plot points using a table of values.

8. Connect the points with a line and add an arrow at the right end of the line to show that the line of the equation continues in that direction. Be sure to write the equation of the line next to the line on the graph.

continued

U2-73





Example 2

A taxi company in Kansas City charges $2.50 per ride plus $2 for every mile driven. Write and graph the equation that models the cost of a taxi ride. Use the slope and the y-intercept to draw the graph.

Example 3

Miranda gets paid $300 each week to deliver groceries. She also earns 5% commission on the total cost of each order she delivers. Write an equation that represents her weekly pay and then graph the equation.

Example 4

The velocity of a ball thrown directly upward can be modeled with the following equation: v = –gt = –gt = – + gt + gt v0, where v is the velocity, g is the acceleration due to gravity, g is the acceleration due to gravity, g t is the elapsed time, and t is the elapsed time, and t v0 is the initial velocity at time 0. If the acceleration due to gravity is equal to 32 feet per second per second, and the initial velocity of the ball is 96 feet per second, what is the equation that represents the velocity of the ball? Graph the equation.

Example 5

A Boeing 747 starts out a long flight with about 57,260 gallons of fuel in its tank. The airplane uses an average of 5 gallons of fuel per mile. Write an equation that models the amount of fuel remaining in the tank over the course of the flight. Graph the equation using a graphing calculator, and then draw the resulting graph on graph paper.

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Name: Date:



Problem-Based Task 2.8: Phone Card Fine PrintProblem-Based Task 2.8: Phone Card Fine PrintWrite and graph the equation that models the following scenario.

You can buy a 6-hour phone card for $5, but the fine print says that each minute you talk actually costs you 1.5 minutes of time. What is the equation that models the number of minutes left on the card compared with the number of minutes you actually talked? What is the graph of this equation? this equation?

What is the equation that

models the number equation that

models the number equation that

of minutes left on the card compared with the number of minutes you actually talked?

What is the graph of this equation?

S MP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

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Practice 2.8: Graphing Linear Equations in Two VariablesPractice 2.8: Graphing Linear Equations in Two VariablesFor problems 1 and 2, graph each equation on graph paper.

1. y = x + 2

2. y x1

32= +

For probl ems 3–10, use the given information to write an equation, then graph the equation on graph paper.

3. A gear on a machine turns at a rate of 2 revolutions per second. Let x represent time in seconds and let y represent the number of revolutions. What is the equation that models the number of revolutions over time?

4. The relationship between degrees Celsius and degrees Fahrenheit is linear. To convert a temperature

from degrees Celsius to degrees Fahrenheit, multiply the temperature by a rate of 9

5 a nd add 32.

What is the equation that models the conversion from degrees Celsius to degrees Fahrenheit?

5. A cab co mpany charges an initial rate of $2.50 for a ride, plus $0.40 for each mile driven. What is the equation that models the total fee for using this cab company?

6. Matthew receives a base weekly salary of $300 plus a commission of $50 for each vacuum he sells. What is the equation that models his weekly earnings?

7. A water company charges a monthly fee of $6.70 plus a usage fee of $2.60 per 1,000 gallons used. What is the equation that models the water company’s total fees?

8. Maddie borrowed $1,250 from a friend to buy a new TV. Her friend doesn’t charge any interest, and Maddie makes $40 payments each month. What is the equation that models the money Maddie owes?

9. A company started with 3 employees and after 8 months grew to 19. The growth was steady. What is the equation that models the growth of the company’s employees?

10. You and some friends are hiking the Appalachian Trail. You started out with 70 pounds of food for the group, and the group eats about 8 pounds of food each day. What is the equation that models the food you have left?

AA

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10. A small newspaper company is downsizing and has lost employees at a steady rate. Twelve months ago they had 65 employees, and now they have 29. What is the equation that models the loss of employees over time?

Practice 2.8: Graphing Linear Equations in Two VariablesFor problems 1–3, graph each equation on graph paper.

1. y = –x = –x = – – 2

2. y = –x = –x = – + 2

3. y x1

24= +

For probl ems 4–10, use the given information to write an equation, then graph the equation on graph paper.

4. A gear on a machine turns at a rate of 1

2 revolution per second. Let x represent time in seconds

and let y represent the number of revolutions. What is the equation that models the number of

revolutions over time?

B

5. The formula for converting temperature from degrees Fahrenheit to degrees Celsius is linear.

To convert from Fahrenheit to Celsius, subtract 32 from the Fahrenheit temperature and then

multiply by a ra te of 5

9. What is the equation that models the conversion of degrees Fahrenheit to

degrees Celsius?

6. A limousine company charges an initial ra te of $50, plus an hourly rate of $75. What is the equation that models the fee for hiring this limousine company?

7. Angela receives a base weekly salary of $100 plus a commission of $65 for each computer she installs. What is the equation that models her weekly pay?

8. A cable company charges a monthly fee of $59 plus $8 for each on-demand movie watched. What is the equation that models the company’s total fees?

9. Garrett borrowed $500 from his aunt. She doesn’t charge any interest, and he makes $15 payments each month. What is the equation that models the amount Garrett owes?


2.9

Date:Name:UNIT 2 • LINEAR FUNCTIONS A–REI.12Lesson 2.9: Solving Linear Inequalities in Two Variables

Warm-Up 2.9Read the scenario and use the information to complete the problems that follow.

Sanibel wants to sell wallets that she makes out of duct tape at the farmer’s market. She bought $20 worth of tape to get started. To pay for both the tape and the time she spends making the wallets, she plans to charge $2.50 for each wallet she sells.

1. Let x represent the number of wallets that Sanibel sells, and let y represent her profit. Write an equation in sl ope-intercept form to show Sanibel’s profit for the wallets she sells.

2. Graph the equation and explain what the graph shows.

3. What happens if Sanibel decides to charge $4 per wallet? Write a new equation in sl ope-intercept form to represent her profit, and graph the equation. Explain what the graph shows.

4. The graph of each of these equations is a line. However, Sanibel is selling entire wallets, not parts of wallets. Explain how you could modify the graphs to better represent the situations.

Lesson 2.9: Solving Linear Inequalities in Two Variables


2.92.9

UNIT 2 • LINEAR FUNCTIONS A–REI.12Lesson 2.9: Solving Linear Inequalities in Two Variables

Name: Date:

For problems 1–7, graph the solution to each inequality.

1.

2.

3.

4.

Scaffolded Practice 2.9: Solving Linear Inequalities in Two Variables

continued


2.9


5.

6.

continued


2.92.9


Name: Date:

7.


8. Emma has at most 2 hours to wash her car and clean her windows. What inequality represents the amount of time Emma has to complete these two tasks? What is the graph of the solution set?

continued


2.9


9. Peter takes 5 minutes to make a sandwich and 10 minutes to make dumplings. If he plans to spend no more than 1 hour making food, what inequality represents the number of sandwiches and dumplings Peter can make? What is the graph of the solution set?

10. Holden has to read his textbooks for history and for literature. If he wants to read at least 100 pages today, what inequality represents the number of pages of history and literature he wants to read? What is the graph of the solutions set?


2.92.9


Name: Date:

continued


Graph the solution to the following inequality.

y > x + 3

1. Graph the linear equation that represents the boundary line.

2 4 6 8 10

10

1 3 5 7 9

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

2. Pick a test point on one side of the line and substitute the coordinates of the point into the inequality.

3. Shade the appropriate half plane.


2.9


Example 2

Graph the solution to the following inequality.

y ≤ –3x + 7

Example 3

A company that manufactures MP3 players needs to hire more workers to keep up with an increase in orders. Some workers will be assembling the players, and others will be packaging them. The company can hire no more than 15 new employees. Write and graph an inequality that represents the number of new workers who can be hired.


2.92.9


Name: Date:

Problem-Based Task 2.9: CupcakesThe class president has asked you to make your delicious cupcakes for the next student council fund-raiser. With the fund-raiser fast approaching, you have asked your friends to help you out. Some friends will frost the cupcakes, and others will decorate the cupcakes. At most, 5 friends have agreed to help. Write and graph an inequality that describes the number of friends who can be assigned to each task if there are at most 5 friends available.

Write and graph an inequality

that describes an inequality

that describes an inequality

the number of friends who can be assigned to each task if there are assigned to each task if there are assigned to each

at most 5 friends available.

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓


2.9


Practice 2.9: Solving Linear Inequalities in Two VariablesFor problems 1–7, graph the s olution to each inequality.

1. y > 2x

2. y > –3x + 1

3. y < –x < –x < – + 3

4. y x1

44≤ −

5. 3x + y ≥ 5

6. 2x – y < 1

7. x > 4


8. Gisele runs a company that makes tablet computers. Each tablet requires an employee to assemble it and an employee to test it. There are 25 employees or fewer available, depending on who is out sick or on vacation. Write an inequality that represents the number of employees Gisele has available to do the work and th en graph the solution set.

9. At Binh’s Greenhouse in early spring, there are many greenhouse plants to repot and many outside plants to water. If it takes Binh 5 minutes to repot each plant and 2 minutes to water each plant, and he has at most 2 hours before the greenhouse opens, what inequality represents the time Binh has to repot and water the plants before the greenhouse opens? What is the graph of the solution set?

10. Toya is training for a triathlon and wants to bike and run on the same day. She has less than 3 hours to spend on her workouts. What inequality represents the time Toya has to bike and run? What is the graph of the solution set?

AA


2.92.9


Name: Date:

BPractice 2.9: Solving Linear Inequalities in Two VariablesFor problems 1–7, gra ph the solution to each inequality.

1. y < 3x – 2

2. y > x – 4

3. y < 2x – 4

4. y ≤ x

5. 2x + 3y + 3y + 3 ≥ –3

6. 4x + y >3

7. y ≤ 2


8. Adult tickets for the high school musical are $12, and student tickets are $8. The drama club needs to sell at least $3,000 worth of tickets to break even on the production. What is an inequality that represents the number of tickets that need to be sold? What is the graph of the sol ution set?

9. Rowan needs to gather pledges for his walk-a-thon to benefit cancer research. People can pledge either a flat donation or a sponsored donation, which will earn him a certain r ate per mile walked. Rowan has a goal of getting more than 200 pledges total. What is an inequality that represents the number of pledges Rowan wants? What is the graph of the solution set?

10. Lisette has 45 minutes or less to complete her homework. She must study for her biology quiz and finish her math homework. What inequality represents the time she has to complete these two tasks? What is the graph of the solution set?

U2-87


Lesson 2.10: Key Features of Linear Functions


Warm-Up 2.10An airplane 50,000 feet above the ground begins to descend for landing. After 5 minutes, the plane is at 40,000 feet.

1. Write two ordered pairs that represent the height of the plane after the given minutes.

2. Write an equation that represents the rate at which the plane is descending.

3. How long does the plane take to land?


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Name: Date:

© Walch Education© Walch Education© Walch Education© Walch EducationNorth Carolina Math 1North Carolina Math 1North Carolina Math 1North Carolina Math 1 Custom Student WorkbookCustom Student WorkbookCustom Student WorkbookCustom Student Workbook2.102.102.102.10

For problems 1–7, find the requested key features.

1. the slope and y-intercept of y = 13x – 3

2. the slope and x-intercept of y – 2 = 8x

3. the slope and y-intercept of 5x + y = 11

4. the slope and x-intercept of 3y-intercept of 3y-intercept of 3 = 2x – 9

5. the domain and range of y = –12x + 4

6. the domain and range of y = 7x – 8

7. the maximum and minimum of y = –3x – 15

continued

Scaffolded Practice 2.10: Key Features of Linear Functions

U2-89




For problem 8, graph the relationship between the given quantities, then use the slope of the line to answer the question.

8. Kutter ordered the same gift for several of his friends at an online store. The gift cost $4, and he paid a flat fee of $9.99 for shipping. Graph the relationship between the number of gifts and the total cost. Then determine how many gifts Kutter bought if his total cost was $49.99. Assume there is no sales tax.

For problems 9 and 10, read the scenario and use the information to answer the questions.

9. A customer at a copy shop has $6.00 remaining on a prepaid card. Black-and-white copies cost $0.12 each, and color copies cost $0.20 each. The equation 12x + 20y + 20y + 20 = 600 models this situation, where x is the number of black-and-white copies and y is the number of color copies the customer can make by using the card. What is the maximum number of color copies the customer can make?

10. The equation y = –15x + 180 models the number of gallons of water, y, in a reef tank x minutes after it has started being drained. What intercept of the graph gives the number of minutes it will take the reef tank to drain? According to the intercept, how many minutes will it take to empty the reef tank?

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Name: Date:


Guided Practice 2.10 Example 1

The following graph illustrates the height of a 10-inch candle that burns at a rate of 2 inches per hour. Write a linear equation that represents the height of the candle. Identify the x- and y-intercepts and the practical domain, then explain their meaning in the context of the problem. How would the graph change if the candle were 13 inches tall? How would the graph change if the candle were 10 inches but burned at a rate of 3 inches per hour?

5

10(0, 10)

15

0

y

x

105

(5, 0)

1. Pick two points on the line and calculate the slope.

2. Write the equation of the line in slope-intercept form.

3. Determine the x-intercept and y-intercept.

4. Determine the practical domain.

5. Create a second equation to represent a candle that begins burning when it is 13 inches tall. The candle still burns at a rate of 2 inches per hour.

6. Graph both equations.

7. Create a second equation to represent a candle that begins burning when it is 10 inches tall, but that burns at a rate of 3 inches per hour.

8. Graph both equations.continued

U2-91




Exa mple 2

The following table shows the distance that Ava traveled while riding her bike around town.

Seconds 18 30 42 54 66Distance traveled (feet) 120 200 280 360 440

If the data in the table represents a linear function, explain the meaning of the slope.

Exa mple 3

Use the following linear functions to calculate the distance between the y-intercepts of f-intercepts of f-intercepts of (f(f x(x( ) and g(x(x( ).

Function 1: f(f(f x(x( ) = 2x – 3 Function 2:

x g(g(g x)x)x–3 –4–1 22 114 17

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Name: Date:


Basketball players are often drafted into the NBA based on their college performance, but they earn future contracts (worth big money) based on their NBA performance. Suppose two players are drafted out of college, and that their per-game points average increases throughout their first 10 seasons based on the following linear functions:

• Player A: f(f(f x(x( ) = 1.7x + 7.4

• Player B:

Season 2 4 7 9 10Points per game 5.8 11 18.8 24 26.6

Compare Player A and Player B. Who had the higher scoring average in his first season? Whose points increased by more each season? Who had a higher scoring average in season 6? Overall, which player has a better career? Defend your answers with mathematical reasoning.

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

Pro blem-Based Task 2.10: Basketball Scoring Averages

Overall, which player has a

better career?

U2-93




Use the function ( )3

42=

−+f x x to complete problems 1–5.

1. What are the domain and range of the function? Is there a maximum or minimum? Explain.

2. What is the rate of change of the function? Is the rate of change increasing or decreasing? Explain.

3. What is the y-intercept of the function? What is the x-intercept of the function?

4. Describe how you would use the key features in problems 1–3 to graph the function.

5. Graph the function ( )3

42=

−+f x x . Does the graph match the description given in problem 4?

Practice 2.10: Key Features of Linear Functions A

continued

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Name: Date:


Use the following information to complete problems 6–10.

A full bathtub is draining. After 5 minutes, there are 25 gallons of water left to drain. After 8 minutes, there are 10 gallons left to drain.

6. Write the equation for this situation. What would be the reasonable domain and range in the context of the problem? Is there a maximum and/or minimum? Explain.

7. What is the rate of change? What does it mean in the context of this problem?

8. Is this function increasing or decreasing? Explain.

9. What is the x-intercept? What is the y-intercept? Explain the meaning of each in the context of the problem.

10. Sketch and describe the graph using the key features in problems 6–9.

U2-95




Use the function ( )2

34=

−+f x x to complete problems 1–5.

1. What are the domain and range of the function? Is there a maximum or minimum? Explain.

2. What is the rate of change of the function? Is the rate of change increasing or decreasing? Explain.

3. What is the y-intercept of the function? What is the x-intercept of the function?

4. Describe how you would use the key features in problems 1–3 to graph the function.

5. Graph the function ( )2

34=

−+f x x . Does the graph match the description given in

problem 4?

Practice 2.10: Key Features of Linear Functions B

continued

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Name: Date:


Use the following information to complete problems 6–10.

A full oil tank is draining oil at a constant rate. After 3 minutes of draining, the tank has 440 liters of oil remaining. After draining for 20 minutes, the tank has 100 liters of oil remaining.

6. Write the equation for this situation. What would be the reasonable domain and range in the context of the problem? Is there a maximum and/or minimum? Explain.

7. What is the rate of change? What does it mean in the context of this problem?

8. Is this function increasing or decreasing? Explain.

9. What is the x-intercept? What is the y-intercept? Explain the meaning of each in the context of the problem.

10. Sketch and describe the graph using the key features from problems 6–9.

U2-97


Lesson 2.11: Graphing Linear Functions



Warm-Up 2.11Geocaching is an outdoor activity that is much like a treasure hunt. Clues to the location of each treasure, or cache, are given for seekers to discover the hidden location. The following coordinates represent the locations of several caches.

A (–3, 5) D (–7, –2)

B (2, 1) E (0, 9)

C (–6, 0) F (3, 0)

1. Use graph paper to graph each point on the same coordinate plane.

2. Identify the quadrant in which each point lies. If a point does not lie in a quadrant, identify if it is an x-intercept or a y-intercept.


U2-98



Name: Date:



For problems 1–7, identify the x- and y-intercepts. Then, graph the function.

1. f(f(f x(x( ) = –2x + 3

x

y

2. f(f(f x(x( ) = 3x – 1

x

y

continued

Scaffolded Practice 2.11: Graphing Linear Functions

U2-99





3. f(f(f x(x( ) = –x) = –x) = – + 6

x

y

4. f(f(f x(x( ) = 4x + 4

x

y

continued

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Name: Date:



5. f(f(f x(x( ) = x – 5

x

y

6. f(f(f x(x( ) = 5x – 8

x

y

continued

U2-101





7. f(f(f x(x( ) = 4

x

y

For problems 8–10, graph the function defined by the given table of values and identify the x- and y-intercepts.

8.

x f(f(f x)x)x

–2 4

0 2

2 0

4 –2 x

y

continued

U2-102



Name: Date:



9.

x f(f(f x)x)x

7 3

3 0

–1 –3

–5 –6 x

y

10.

x f(f(f x)x)x

4 3

2 1

0 –1

–2 –3 x

y

U2-103





continued


Given the function f x x( )3

26= − , use a table of values to graph and identify the x- and y-intercepts.

1. Create a table of values.

2. Plot two points from the table.

2 4 6 8 10

10

1 3 5 7 9

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

y

x

–10

–1–2–3–4–5–6–7–8–9–10

3. Draw the line connecting the two points. Be sure to extend the line so that it intersects both the x- and y-axes.

4. Identify the x-intercept.

5. Identify the y-intercept.

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Name: Date:



Example 2


52=− + , use the slope and y-intercept to graph the function. Then, identify

the x-intercept of the function.

Example 3


34=− + , solve for the x- and y-intercepts. Use the intercepts to graph

the function.

Example 4


820=− + , graph the function using technology. Identify the intercepts.

U2-105





Problem-Based Task 2.11: Fund-raising ConcertYou are helping to organize a fund-raising concert for your community awareness group. Concer t tickets will sell for $20 each in advance, and for $30 each on the day of the show. Your club’s goal is to raise $6,000. Wri te an equation in standard form for the function that represents this scenario. Draw the graph of the function that represents how many of each type of ticket you will need to sell. If you sell only advance tickets, how many tickets do you need to sell? Where do you find this on your graph? If you sell only same-day tickets, how many tickets do you need to sell? Where do you find this on your graph?

If you sell only advance tickets,

how many tickets do you need to

sell? If you sell only same-day

tickets, how many tickets do you need

to sell?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

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Name: Date:



continued

Practice 2.11: Graphing Linear FunctionsUse what you know about linear functions to complete the following problems.

1. Given the function f x x( )4

34=− + , use the slope and y-intercept to graph the function. Identify

the x- and y-intercepts.


72= + , use the slope and y-intercept to graph the function. Identify


3. Given the table of values, graph the function and identify the x- and y-intercepts.

x f(f(f x)x)x–3 100 53 06 –5


x f(f(f x)x)x–14 –2–7 00 27 4


32=− + , solve for the x- and y-intercepts. Use the intercepts to graph

the function.


45= − , solve for the x- and y-intercepts. Use the intercepts to graph

the function.

AA

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7. Kaylee is selling candles to raise money for her lacrosse team. The large candles sell for $25 each and the small candles sell for $10 each. She needs to raise $600. Write a function to represent how many of each type of candle Kaylee needs to sell. Draw the graph of the function. If she sells only large candles, how many candles does she need to sell? If she sells only small candles, how many candles does she need to sell?

8. Jerome knits scarves and hats for charity. Scarves require 4 skeins of yarn and hats require 2 skeins of yarn. Jerome has 24 skeins of yarn. Write a function to represent the combination of scarves and hats he can knit. Draw the graph of the function. If Jerome knits only scarves, how many can he knit? If he knits only hats, how many can he knit?

9. A farmer raises goats and cows. Each goat requires 400 square feet of grazing area and each cow requires 1,200 square feet of grazing area. The farmer has 36,000 square feet for grazing area. Write a function to represent the combination of goats and cows that the farmer can raise. Use technology to graph the function. If the farmer raises only cows, how many can he raise? If the farmer raises only goats, how many can he raise?

10. The graph of a function is shown. Write a scenario that could be represented by the graph.

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Name: Date:



Practice 2.11: Graphing Linear FunctionsUse what you know about linear functions to complete the following problems.


34=− − , use the slope and y-intercept to graph the function. Identify


2. Given the function f(f(f x(x( ) = –3x + 9, use the slope and y-intercept to graph the function. Identify the x- and y-intercepts.


x f(f(f x)x)x–7 –6–5 0–3 60 15


x f(f(f x)x)x–2 –80 –42 03 2


37= + , solve for the x- and y-intercepts. Use the intercepts to graph

the function.

6. Given the function f(f(f x(x( ) = –8x – 16, solve for the x- and y-intercepts. Use the intercepts to graph the function.

B

continued

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7. Chris is collecting cans and bottles to raise money for his baseball team. He gets $6 for each bag of bottles collected and $5 for each bag of cans collected. Chris needs to collect $30 worth of cans and bottles. Write a function to represent how many bags of cans and bottles Chris needs to collect. Draw the graph of the function. If Chr is collects only bottles, how many bags of bottles does he need to collect? If he collects only cans, how many bags of cans does he need to collect?

8. Ashlyn makes leather handbags and belts to sell online. The belts require 50 square inches of leather and the handbags require 350 square inches of leather. Ashlyn has 700 square inches of leather available. Write a function to represent the combination of belts and handbags she can make. Draw the graph of the function. If Ashlyn makes only belts, how many can she make? If she makes only handbags, how many can she make?

9. A farmer grows corn and wheat. Each acre of corn requires takes 40 hours to plant. Each acre of wheat takes 120 hours to plant. The farmer has 360 hours to plant the crops. Write a function to represent the combination of wheat and corn that the farmer can plant. Draw the graph of the function. If the farmer grows only corn, how many acres can he plant? If the farmer plants only wheat, how many acres can he plant?

10. The graph of a function is shown. Write a scenario that could be represented by the graph.

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UNIT 2 • LINEAR FUNCTIONS F–IF.9Lesson 2.12: Comparing Linear Functions

Name: Date:



Warm-Up 2.12Cecilia took her first parachute jump lesson last weekend. Her instructor gave her the following graph, which shows her change in altitude in meters during a 5-second interval. Use the graph to complete the problems that follow.

0 1 2 3 4 5

300

600

900

1200

1500

1800

2100

2400

2700

Alti

tude

(met

ers)

Time (seconds)

1. Estimate Cecilia’s average rate of change in altitude in meters per second.

2. What are the domain and range for this function?

Lesson 2.12: Comparing Linear Functions

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Date:Name:UNIT 2 • LINEAR FUNCTIONS F–IF.9Lesson 2.12: Comparing Linear Functions



Compare the properties of the following linear functions.

1. Which function has the greater rate of change? Which function has the greater y-intercept? Explain how you know.

Function A Function B

x f(f(f x) x) x

–2 –1

2 3

3 4

5 6 2 4

–2

–4

–3

–5

–1

4

2

5

3

1

0 1 3 5

y

x

–1–2–3–4–5

2. Which function has the greater rate of change? Which function has the greater y-intercept? Explain how you know.


x f(f(f x) x) x

–6 15

–1 0

2 –9

4 –15 2 4

–2

–4

–3

–5

–1

4

2

5

3

1

0 1 3 5

y

x

–1–2–3–4–5

continued

Scaffolded Practice 2.12: Comparing Linear Functions

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Name: Date:



3. Compare the properties of each function.


= −( )1

36f x x

2 4

–2

–4

–3

–5

–1

4

2

5

3

1

0 1 3 5

y

x

–1–2–3–4–5


Function A Function Bf(f(f x(x( ) = 12x

2 4

–2

–4

–3

–5

–1

4

2

5

3

1

0 1 3 5

y

x

–1–2–3–4–5

continued

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Function A Function BThis table describes the amount of profit in dollars a store makes for the number of computers it sells.

Number of computers (x)x)x Profit (f(f( (f(f x))x))x

0 $0

5 $1,000

8 $1,600

10 $2,000

For each phone the store sells, it makes a profit of $500.


Function A Function BA mining company removed 100 tons of dirt in its first year, and 28 tons every subsequent year.

The function g(x(x( ) = 17 + 56x represents the total number of tons removed by a rival mining company in the same time period, where g(x(x( ) is the number of tons and x is the number of years since the first year.

continued

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Name: Date:




Function A Function BThe following table gives the price in dollars, f(f(f x(x( ), a company charges to rent a surfboard for x hours.

Hours (x)x)x Total cost (fTotal cost (fTotal cost ( (f(f x)) x)) x

2 30

3 35

6 50

8 60

A rival company rents surfboards for $10 initially and then $7 per hour.


Function A Function BThis table represents the remaining amount of debt in thousands of dollars a person has, f(f(f x(x( ), after x months.

Months (x)x)xRemaining debt in

thousands of $ (fthousands of $ (fthousands of $ ( (f(f x)) x)) x

0 6.6

2 5.8

3 5.4

5 4.6

This graph shows the remaining amount of debt in thousands of dollars another person has, g(x(x( ), after x months.

6

4

2

5

3

1

2 4 61 3 5

y

x0

continued

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9. Compare the properties of each function. What do the y-intercept and rate of change mean in each case?

Function A Function BThis table represents the number of apples, f(f(f x(x( ), waiting to be picked in an orchard after x days.

Days (x)x)xApples (f(f( (f(f x)) x)) x

0 49

1 43

4 25

7 7

The function g(x(x( ) = 32 – 4x represents the number of apples, g(x(x( ), remaining to be picked in a different orchard after x days.

10. Compare the properties of each function. What do the y-intercept and rate of change mean in each case?

Function A Function BJohn ran 6 miles last week and plans to run 12 miles each additional week.

This graph represents the total number of miles, g(x(x( ), Grant plans to have run by the end of week x.

2 4

8

6

4

2

1 3

y

x0

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continued


The functions f(f(f x(x( ) and g(x(x( ) are shown. Compare the properties of each.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

f(x)

x g(g(g x)x)x–2 –10–1 –80 –61 –4

1. Identify the rate of change for the first function, f(f(f x(x( ).

2. Identify the rate of change for the second function, g(x(x( ).

3. Identify the y-intercept of the first function, f(f(f x(x( ).

4. Identify the y-intercept of the second function, g(x(x( ).


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continued

Example 2

Your employer has offered two pay scales for you to choose from. The first option is to receive a base salary of $250 a week plus 15% of the price of any merchandise you sell. The second option is represented in the graph, where x represents the price of the merchandise sold and y represents your weekly salary. Compare the properties of the functions.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Wee

kly

sala

ry ($

)

Total price of merchandise sold ($)

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Example 3

Two airplanes are in flight. The function f(f(f x(x( ) = 400x + 1200 represents the altitude in meters, f(f(f x(x( ), of one airplane after x minutes. The following graph represents the altitude of the second airplane after x minutes. Compare the properties of the functions.

0 1 2 3 4 5 6 7 8 9 10

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Alti

tude

(m)

Time (minutes)

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Problem-Based Task 2.12: Supply and DemandIdeal Electronics is determining the price of the newest tablet to hit the market. In an effort to make the most money and sell the most tablets, Ideal Electronics wants to price the tablet appropriately to the product’s supply and demand. Supply is the number of tablets that are available and demand is the amount that buyers are willing to pay. The relationship between supply and demand often influences the price of products.

Supply is modeled by the linear function f(f(f x(x( ) = 0.3x + 100, where f(f(f x(x( ) represents the price per tablet in dollars and x represents the number of tablets.

Demand is modeled in the following table, where g(x(x( ) represents the price per tablet in dollars and x represents the number of tablets.

x g (x)x)x100 490300 370500 250600 190

Compare the properties of both of the functions described. At what point does the supply of tablets exceed the demand? Explain your reasoning.

At what point does the supply of tablets exceed the

demand?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

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Name: Date:



continued

Practice 2.12: Comparing Linear FunctionsCompare the properties of the linear functions.

1. Which function has a greater rate of change? Which function has the greater y-intercept? Explain how you know.

Function A

x f(f(f x)x)x–4 12–1 02 –123 –16

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)


Function A

x f(f(f x)x)x–8 10 24 2.58 3

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)

AA

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Function A

( )1

43= +f x x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)


Function A

f(f(f x(x( ) = –5x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)

continued

U2-122


Name: Date:




Function A

The following table describes the profit in dollars that a restaurant makes for the number of beverages it sells.

Number of beverages sold (x)x)x

Profit (f(f( (f(f x))x))x

0 025 29.2550 58.5075 87.75

Function B

For each hamburger sold, the same restaurant makes a profit of $0.40.


Function A

A local newspaper began with a circulation of 1,300 readers in its first year. Since then, its circulation has increased by 150 readers per year.

Function B

The function g(x(x( ) = 225x + 950 represents the circulation of another newspaper where g(x(x( ) represents total subscriptions and x represents the number of years since its first year.


Function A

A rental store charges $40 to rent a steam cleaner, plus an additional $4 per hour.

Function B

The following table shows the total cost in dollars to rent a steam cleaner at a different rental store. g(x(x( ) represents the total cost after x hours.

Hours (x)x)x Total cost (g(g( (g(g x))x))x3 464 535 606 67

continued

U2-123





Function A

The table shows the remaining balance in dollars, f(f(f x(x( ), of the cost of car repairs after x months.

Months (x)x)xRemaining

balance (fbalance (fbalance ( (f(f x))x))x0 15601 14302 13003 1170

Function B

The graph shows the remaining balance in dollars, g(x(x( ), of the cost of car repairs after x months.

0 1 2 3 4 5 6 7 8 9 10 11 12

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Rem

aini

ng b

alan

ce

Months

9. Compare the properties of each function. What do the rate of change and y-intercept mean in terms of the scenarios?

Function A

The function f(f(f x(x( ) = 7.5 – 0.25xrepresents the pounds of puppy food remaining, f(f(f x(x( ), when the puppy is fed the same amount each day for x days.

Function B

The table represents the amount in pounds of puppy food remaining, g(x(x( ), when the puppy is fed the same amount each day for x days.

Days (x)x)x Remaining food (g(g( (g(g x))x))x4 95 8.756 8.57 8.25

continued

U2-124


Name: Date:




Function A

Reggie bicycled 15 miles last week and plans to bicycle 20 miles each additional week.

Function B

The graph represents the total number of miles Zac plans to have bicycled by the end of each week.

0 1 2 3 4 5 6 7 8 9 10 11 12

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

Mile

s

Weeks

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Function A

x f(f(f x)x)x–14 –2–7 –30 –47 –5

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

Practice 2.12: Comparing Linear FunctionsCompa re the properties of the linear functions.


Function A

x f(f(f x)x)x–3 –140 –52 15 10

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

B

continued

U2-126


Name: Date:



continued


Function A

( )2

39= +f x x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10


Function A

f(f(f x(x( ) = 3x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

U2-127




continued


Function A

A game store charges $3.50 to rent a video game for one night, plus an additional $2 per day thereafter.

Function B

The table shows the total cost to rent the same game at a different rental store, where fstore, where fstore, where (f(f x(x( ) represents the total cost in dollars after x days.x days.x

x f(f(f x)x)x2 6.003 8.504 11.005 13.50


Function A

The table describes the profit in dollars made on ice creams sold by a street vendor.

Number of ice creams sold (x)x)x

Profit (fProfit (fProfit ( (f(f x))x))x

0 020 4.6040 9.2060 13.80

Function B

For each hot dog sold, the same vendor makes a profit of $0.20.


Function A

The local community magazine began with a circulation of 3,400 subscribers in its first year. Since then, its circulation has increased by 175 subscribers per year.

Function B

The function f(f(f x(x( ) = 95x + 2200 represents the circulation of another magazine in a nearby community, where f(f(f x(x( ) represents total subscriptions and x represents the number of years since it began its circulation.

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Name: Date:



continued


Function A

The following table shows the remaining balance, f(f(f x(x( ), of the cost of pool repairs after x months.

x f(f(f x)x)x0 12002 10504 9006 750

Function B

This graph shows the remaining balance, g(x(x( ), of the cost of pool repairs after x months.

0 1 2 3 4 5 6 7 8 9 10 11

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

850

Cost

of p

ool r

epai

rs

Months

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Function A

The function f(f(f x(x( ) = 12.5 – 0.32xrepresents f(f(f x(x( ), the amount of cat food remaining in pounds when a cat is fed the same amount each day for x days.

Function B

The table represents g(x(x( ), the amount of cat food remaining in pounds when a cat is fed the same amount each day for x days.

x g (x)x)x3 9.044 8.725 8.406 8.08


Function A

Sophie ran 8 miles last week and plans to run 2 miles each additional week.

Function B

The following graph represents Kaelina’s running plan.

0 1 2 3 4 5 6 7 8 9 10

5

10

15

20

25

30

Mile

s

Weeks

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UNIT 2 • LINEAR FUNCTIONS F–BF.1a★

Lesson 2.13: Building Functions from Context

Name: Date:



Warm-Up 2.13For each problem, write an equation to represent the situation and then answer the question.

1. Willem buys 4 mangoes each week, and mango prices vary from week to week. Write an equation that represents the weekly cost of the mangoes. If each mango costs $1 this week, what is the total cost of his mangoes for this week?

2. Kerin drives at a speed of 55 miles per hour on the highway for her job. Write an equation that represents the distance she travels as a function of the time she spends driving. If one day she drives for 6 hours, how many miles did she travel?

3. Mr. Stevens teaches 4 math classes every day. Depending on absences, the number of students in each class varies. Write an equation that represents the number of students Mr. Stevens teaches in a day. If there are 30 stude nts taking each class and one day all of the students were present, how many students did Mr. Stevens teach that day?

4. Jessica reads approximately 12 pages of her novel each hour. Depending on extracurricular activities and homework, the time that Jessica has to read varies. Write an equation that represents the number of pages Jessica reads as a function of the time spent reading. If Jessica read for 3 hours yesterday, approximately how many pages did she read?


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Date:Name:UNIT 2 • LINEAR FUNCTIONS F–BF.1a★




Write an explicit function to represent each pattern.

1. Drew is holding a fund-raiser. He is taking donations from his family members to support a charity. Each family member donates the same amount. The total amounts donated after 1, 2, 3, and 4 family members give money are $20 $30, $40, and $50, respectively.

2. Emily goes on vacation with $350. Each day, she spends the same amount of money. After 1, 2, 3, and 4 days on vacation, she has $308, $266, $224, and $182, respectively.

3. Grandma is picking apples for her apple pies. When she starts picking them, she has 4 apples in her bucket. After 1, 2, 3, and 4 minutes of picking apples, she has 11, 18, 25, and 32 total apples, respectively.

4. Housepainters work together to complete a painting project. One painter can paint 4 square feet in a minute. Two painters can paint 8 square feet in a minute. Three painters can paint 12 square feet in a minute, and four painters can paint 16 square feet in a minute.

continued

Scaffolded Practice 2.13: Building Functions from Context

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Name: Date:



5. Joe puts a cup of soup in the freezer, and records the temperature after each minute. At 0 minutes, the soup starts at 68°F. After 1, 2, and 3 minutes, the temperature is 54°F, 40°F, and 26°F, respectively.

6. Given the diagram that follows, describe the number of squares in Figure x if the pattern continues.x if the pattern continues.x

Figure 1 Figure 2 Figure 3

7. Given the diagram that follows, describe the number of triangles in Figure x if the pattern x if the pattern xcontinues.

Figure 1Figure 1 Figure 3Figure 2

continued

U2-133





8. The number of end points is increasing from figure to figure in the diagram shown. Write an explicit function to find the total number of end points in any figure.


9. An overnight shipping service charges a fixed fee of $12.00, plus an additional fee based on the weight of the item being shipped. The service charges an additional $0.15 for each pound. Find an explicit function to represent the shipping charge for a package of any weight.

10. The value of a car decreases over time. Mario’s car was originally worth $25,000. Each year, his car is worth approximately $1,300 less than the year before. Find an explicit function to represent the value of the car in any year.

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Name: Date:



continued


The starting balance of Anna’s account is $1,250. She takes $30 out of her account each month. How much money is in her account after 1, 2, and 3 months? Find an explicit function to represent the balance in her account at any month.

1. Use the description of the account balance to find the balance after each month.

2. Determine the independent and dependent quantities.

3. Determine if there is a common difference or common ratio that describes the change in the dependent quantity.

4. Use the common difference to write an explicit function.

5. Evaluate the function to verify that it is correct.

U2-135





Example 2

A video arcade charges an entrance fee, then charges a fee per game played. The entrance fee is $5, and each game costs an additional $1. Find the total cost for playing 0, 1, 2, or 3 games. Describe the total cost of playing x games with an explicit function.

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Name: Date:



Problem-Based Task 2.13: Interior Angles in Polygons Problem-Based Task 2.13: Interior Angles in Polygons Julia is studying the sum of the measures of the interior angles in polygons. She creates Julia is studying the sum of the measures of the interior angles in polygons. She creates polygons with 3, 5, 6, 7, and 9 sides, and records the sum of the interior angles in each polygon in the following table.

Number of sides Sum of interior angles, in degrees3 1805 5406 7207 9009 1260

Is the re a relationship between the number of sides of a polygon and the sum of the interior angles? Write a function that can be used to determine the sum of the interior angles of a polygon with any number of sides.with any number of sides.

Is there a relationship

between the number of sides of a

polygon and the sum of the interior

angles?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

U2-137





continued

Practice 2.13: Building Functions from ContextWrite an explicit function to represent each pattern.

1. Mr. Ramos notices a pattern in the number of people attending the weekly student government meetings. For weeks 1, 2, 3, 4, and 5, the number of students attending the meeting was 31, 43, 55, 67, and 79, respectively.

2. Hannah borrows $30 from her parents. Each week, she pays them back the same amount. The total amounts she owes her parents after weeks 0, 1, 2, 3, and 4 are $30, $25, $20, $15, and $10, respectively.

3. Angelo sells cookies in packages, where each package contains the same number of cookies. The total number of cookies he has after 1, 2, 3, 4, and 5 packages are sold are 110, 88, 66, 44, and 22, respectively.

4. Cameron tracks the number of people who read his blog. In weeks 1, 2, 3, 4, and 5, the blog had 100, 150, 200, 250, and 300 visitors, respectively.

5. As a treat, Nia eats a portion of a chocolate bar each day. She eats the same portion of the remaining bar each day. On day 0, the bar of chocolate starts with 32 pieces. After 1 day, 26 pieces remain. After days 2, 3, and 4, there are a total of 20, 14, and 8 pieces remaining.

6. Given the diagram, if the pattern continues, describe the number of sides in Figure x.

Figure 1 Figure 2 Figure 3 Figure 4

AA

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Name: Date:



7. Given the diagram that follows, describe the number of blocks in Figure x if this pattern continues.


8. Brandon sells candy in packages, and each package contains the same number of pieces of candy. The total number of pieces of candy he has after 1, 2, 3, 4, and 5 packages have been sold are 15, 30, 45, 60, and 75, respectively.

9. A hotel charges a room fee per night, plus an additional fee if more than one guest is staying in a room. Good Nights hotel charges $150 per night for a room, plus $25 per guest if more than one guest is staying in a room. Find an explicit function to represent the nightly cost for any number of guests.

10. The population of a city is growing. Each year, the population increases by approximately 5,000 people over the previous year’s population. The population this year is 10,000. Find an explicit function to represent the population of the town in any year. Consider that year 0 is this year.this year.

U2-139





Practice 2.13: Building Functions from ContextWrite an explicit function to represent each pattern.

1. Pedro is holding a fund-raiser. He is taking donations from his friends to support a charity. Each friend donates the same amount. The total amounts donated after 1, 2, 3, and 4 friends give money are $15, $30, $45, and $60, respectively.

2. Diana goes on vacation with $260. Each day, she spends the same amount of money. After 1, 2, 3, and 4 days on vacation, she has $242, $224, $206, and $188, respectively.

3. Gemma is picking blueberries. When she starts picking them, she has 7 berries in her bucket. After 1, 2, 3, and 4 minutes of picking berries, she has 16, 25, 34, and 43 total berries, respectively.

4. Housepainters work together to complete a painting project. One painter can paint 6 square feet in a minute. Two painters can paint 16 square feet in a minute. Three painters can paint 26 square feet in a minute, and four painters can paint 36 square feet in a minute.

5. Isaac records the temperature of a cup of water each minute. At 0 minutes, the water starts at 60°F. After 1, 2, and 3 minutes, the temperature is 54°F, 48°F, and 42°F, respectively.

6. Given the diagram that follows, describe the number of triangles in Figure x if the pattern continues.

Figure 1 Figure 2 Figure 3 Figure 4

B

continued

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Name: Date:



7. Given the diagram that follows, describe the number of squares in Figure x if the pattern continues.


8. Mario sells pencils in packages, and each package contains the same number of pencils. The total number of pencils he has after selling 1, 2, 3, 4, and 5 packages are 12, 24, 36, 48, and 60, respectively.

9. An overnight shipping service charges a fixed fee of $10.00, plus an additional fee based on the weight of the item being shipped. The service charges an additional $0.25 for each pound. Find an explicit function to represent the shipping charge for a package of any weight.

10. The value of a car decreases over time. Mario’s car was originally worth $15,000. Each year, his car is worth approximately $1,000 less than the year before. Find an explicit function to represent the value of the car in any year.

U2-141

Date:Name:UNIT 2 • LINEAR FUNCTIONS F–BF.2★

Lesson 2.14: Arithmetic Sequences



Warm-Up 2.14Just as Julian finished his homework, he went to feed his fish and accidentally dropped his homework in the fish tank. He was working on patterns of numbers, and the last number in four of the patterns was washed away by the water in the fish tank. Julian needs help recreating his patterns. For each problem, describe the pattern and determine the next number in the list.

1. 1, 3, 5, 7, …

2. 13, 18, 23, 28, …

3. 7, 4, 1, –2, –5, …

4. –22, –15, –8, –1, …


U2-142

UNIT 2 • LINEAR FUNCTIONS F–BF.2★


Name: Date:



For problems 1–4, find the common difference and write the explicit formula for the nth term of each arithmetic sequence.

1. 3, 9, 15, 21, ...

2. 42, 38, 34, 30, ...

3. –8, –5, –2, 1, ...

4. 2, 0.5, –1, –2.5, ...

For problems 5–8, find the first five terms of the given arithmetic sequence.

5. an = 5 – 3(n – 1)

6. an = an – 1 + 4, a1 = –5

7. an = 2 + an – 1, a1 = 0

8. an = 1 + 3n, a1 = 4

continued

Scaffolded Practice 2.14: Arithmetic Sequences

U2-143





Use the given information to complete problems 9 and 10.

9. Write the explicit formula for the nth term of the arithmetic sequence if the common difference d = 2, with d = 2, with d a1 = 3.

10. Write the explicit formula for the nth term of the arithmetic sequence if the common difference d = –3, with d = –3, with d a1 = 1.

U2-144



Name: Date:



continued


Find the common difference, write the explicit formula, and find the tenth term for the following arithmetic sequence.

3, 9, 15, 21, …

1. Find the common difference by subtracting two successive terms.

2. Confirm that the difference is the same between each remaining pair of consecutive terms.

3. Identify the first term, a1.

4. Write the explicit formula.

5. Simplify the explicit formula.

6. To find the tenth term, substitute 10 for n in the explicit formula.

U2-145





Example 2

Write a linear function that corresponds to the following arithmetic sequence.

8, 1, –6, –13, …

Example 3

An arithmetic sequence is defined recursively by the formula an = an – 1 + 5, with a1 = 29. Find the first 5 terms of the sequence, write an explicit formula to represent the sequence, and find the 15th term.

U2-146



Name: Date:



Problem-Based Task 2.14: New TabletYou are saving money so you can buy a new tablet computer. The tablet costs $450. Your grandparents gave you $50 for your birthday. Before that $50 gift, you didn’t have any money saved. You have a babysitting job and plan to save $10 each week. Write an arithmetic sequence to represent the amount of money you will have at the end of each week. In how many weeks will you have saved enough money to buy the tablet?

In how many weeks will you have saved enough money to buy the tablet?

SMP1 ✓ 2 ✓3 ✓ 4 ✓5 ✓ 6 ✓7 ✓ 8 ✓

U2-147





Practice 2.14: Arithmetic SequencesFor problems 1–4, find the common difference and write the explicit formula for the nth term of each arithmetic sequence.

1. 27, 31, 35, 39, …

2. 4, –3, –10, –17, …

3. –101, –87, –73, –59, …

4. 1

2,

5

2,

9

2,13

2,...


5. Find the first five terms of the arithmetic sequence defined as follows:

an = an – 1 + 2.7; a1 = 3.2


an = an – 1 – 22; a1 = 18

7. You have read 25 pages of a book. You plan to read an additional 10 pages each night. Write the explicit formula to represent the number of pages you will read after n nights.

8. You are going on vacation. You have $105 to take with you. You expect to spend $15 each day. You want to have $30 remaining at the end of the vacation. Write an explicit formula to represent this scenario. For how many days can you spend $15 each day?

9. A bicyclist is training for a race. On the first day of training, she rides 12 miles. She increases the distance she rides by 3 miles each day. Write an explicit formula to represent this scenario. How many miles will the bicyclist ride on her ninth day of training?

10. Sofie needs to complete community service hours for her service club. She needs to complete 150 hours to earn a merit badge. Sofie has already completed 65 hours. Write an explicit formula to represent this scenario. If she volunteers 5 hours each week, in how many weeks will she have completed the hours to earn the merit badge?

AA

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Name: Date:



Practice 2.14: Arithmetic SequencesFor problems 1–4, find the common difference and write the explicit formula for the nth term of each arithmetic sequence.

1. 4.2, 6, 7.8, 9.6, …

2. 11, 3, –5, –13, …

3. –237, –194, –151, –108, …

4. 5

3,

8

3,11

3,14

3,...

B


an = an – 1 – 31; a1 = 52

7. You have read 15 pages of a book. You plan to read an additional 12 pages each night. Write the explicit formula to represent the number of pages you will read after n nights.

8. You have $53 in your lunch account. You spend $3 each day for lunch. You need to have $5 remaining at the end the month to keep your account open. Write an explicit formula to represent this scenario. For how many days can you buy lunch?

9. Jaden is starting a wellness plan. Walking each day is part of her plan. She begins by walking for 1

2 hour on the first day. She plans to increase by

1

4 hour each day. Write an explicit formula to

represent this scenario. After how many days will Jaden be walking for 13

4 hours?



an = an – 1 + 0.6; a1 = 12.3

10. Augie is saving to buy a new scooter. He has $78 in his account. He delivers newspapers and plans to save $14 each week. Write an explicit formula to represent this scenario. How much money will Augie have at the end of 11 weeks?


Station Activities Set 1: Comparing Linear ModelsStation Activities Set 1: Comparing Linear Models

Date:Name:UNIT 2 • LINEAR FUNCTIONS A–CED.2★, A–REI.10, F–IF.7★

Station Activities Set 1: Comparing Linear Models

continued

Station 1You will be given a ruler and graph paper. Work together to analyze data from the real-world situation described, then, as a group, answer the questions.

Keri i s going to get a new cell phone and she has to choose between two cell phone companies. 5-Bars Phone Company charges $50 per month. The company charges an additional $0.60 per minute if a customer uses more than the monthly number of minutes included in the plan. Stellular Phone Company charges $70 per month. This company charges an additional $0.20 per minute if a customer uses more than the monthly number of minutes included in the plan. Both companies’ plans include the same number of minutes each month.

Let x represent the minutes used that exceeded the plan. Let y represent the cost of the plan.

1. Write an equation that represents the monthly cost of 5-Bars Phone Company’s plan.

Complete the table by selecting values for x and calculating y.

Minutes (x)x)xCost in $ (yCost in $ (yCost in $ ( )y)y

Use your graph paper to graph the ordered pairs. Use your ruler to draw a straight line through the po ints and complete the graph.

2. Write an equation that represents the monthly cost of Stellular Phone Company’s plan.

Complete the table by selecting values for x and calculating y.

Minutes (x)x)xCost in $ (yCost in $ (yCost in $ ( )y)y

O n the same graph you created for problem 1, plot the ordered pairs. Use your ruler to draw a straight line through the poin ts and complete the graph.



Name: Date:UNIT 2 • LINEAR FUNCTIONS A–CED.2★, A–REI.10, F–IF.7★


Use your graph from problems 1 and 2 to answer the following questions.

3. Which plan should Keri choose if she uses 30 minutes of extra time each month? Explain.

4. Which plan should Keri choose if she uses 80 minutes of extra time each month? Explain.

5. At what number of extra minutes per month would it not matter which phone plan Keri chooses since the cost would be the same? Explain.

6. If the equations for the two cell phone plans were solved as a system of equations, what would be the solution? Explain.





Station 2The equation y = 0.6x represents the number of calories (y represents the number of calories (y represents the number of calories ( ) that a runner burns per mile based on the runner’s body weight of x pounds.

Body weight (pounds)

Calories Burned per Mile

Calo

ries

burn

ed

90 100 110 120 130 140 150 160

90

85

80

75

70

65

60

55

50

45

40

0

x

y100

95

For each weight given, use the graph to find the number of calories burned per mile.

1. 100 pounds

2. 115 pounds

3. 135 pounds

For each given number of calories burned per mile, use the graph to find the matching weight of the person.

4. 75 calories burned

5. 90 calories burned

6. If you didn’t know the equation of this graph, how could you use the graph to find the equation of the line? Explain.





continued

Station 3NOAA Satellite and Information Service created the following graph, which depicts the U.S. National Summary of the temperature in January from 1999–2009.

1999 2000 2003 2008

32.0

30.0

2001 2002 2004 2005 2006 2007 2009

30.531.031.5

32.533.033.534.0

35.035.5

34.5

36.036.537.037.538.0

39.5

38.539.0

40.0

Year

Tem

pera

ture

National Summary of January Temperatures1999–2009

TempertureAverageSource: National Oceanic and Atmospheric Administration

1. Between which consecutive years did the United States see the greatest increase in average temperature change in January?

2. What strategy did you use to answer problem 1?





3. Between which consecutive years did the United States see the greatest decrease in average temperature change in January?

4. What strategy did you use to answer problem 3?

5. Between which consecutive years was the temperature change represented as a negative slope? Explain.

6. Between which consecutive years was the temperature change represented as a positive slope? Explain.Explain.





continued

Station 4You will work with a linear function at this station.

Use the given linear function for the following problems.

f(f(f x(x( ) = x + 3

1. Create a table of values for the function.

x f(f(f x)x)x

2. Find the x- and y-intercepts.

3. Graph the function on the coordinate plane.

2 4 6 8 10

10

–2

–4

–6

–8

–9

–3

–5

–7

–1

8

6

4

2

7

5

3

1

0

9

1 3 5 7 9

y

x

–10

–1–2–3–4–5–6–7–8–9–10





4. Looking at the equation, what is a defini ng characteristic of a linear function?

5. Looking at the table of values, what is a defining characteristic of a linear function’s table of values?

6. Looking at the graph, what is a defining characteristic of a linear function’s graph?Looking at the graph, what is a defining characteristic of a linear function’s graph?


Station Activities Set 2: Relations Versus Functions/Domain and RangeStation Activities Set 2: Relations Versus Functions/Domain and Range

Name: Date:UNIT 2 • LINEAR FUNCTIONS F–BF.1a★, F–IF.1, F–IF.2Station Activities Set 2: Relations Versus Functions/Domain and Range

Station 1You will be given eight index cards with the following functions and function values written on them:

f(f(f x(x( ) = 2x; f(f(f x(x( ) = –3x + 7; f(f(f x(x( ) = x2; f x x( ) =23

; f(3) = 6; f(3) = 6; f f(3) = 9; f(3) = 9; f f(3) = –2; f(3) = –2; f f(3) = 2f(3) = 2f

1. Work together to match each function with its corresponding function value. Write your matches in the space provided.

Evaluate each function for the given expression. Show your work.

2. Let f(f(f x(x( ) = x + 5. What is f(f(f x (x ( + 3)?

3. Let f(f(f t) = t) = t t 2. What is f(f(f t – 4)?

4. Let f s s( ) =15

. What is f(f(f s + 4)?



Date:Name:UNIT 2 • LINEAR FUNCTIONS F–BF.1a★, F–IF.1, F–IF.2Station Activities Set 2: Relations Versus Functions/Domain and Range

continued

Station 2You will be given a ruler and graph paper. As a group, use your ruler to determine whether each of the following relations is a function. Beside each graph, write your answer and reasoning.

1.

1 2 3 4 5

5

–1

–2

–3

–4

4

3

2

1

0

y

y = 2x

x

–5

–1–2–3–4–5

2.

1 2 3 4 5

5

–1

–2

–3

–4

4

3

2

1

0

y

x

–5

–1–2–3–4–5

x2 + y2 = 4




3.

– 10 – 8 – 6 – 4 – 2 20 4 6 8 10

x

y10

8

6

4

2

– 2

– 4

– 6

– 8

– 10

y = 2x + 1

How did you use your ruler to determine whether each relation was a function?

4. Use your ruler and graph paper to sketch a function. Use the vertical line test to verify that it is a function.

Determine whether each of the following relations is a function. Explain your answer.

5. {(2, 5), (3, 1), (1, 4), (3, 6)}

6. {(1, 1), (2, 1), (3, 2)}



Date:Name:UNIT 2 • LINEAR FUNCTIONS F–BF.1a★, F–IF.1, F–IF.2Station Activities Set 2: Relations Versus Functions/Domain and Range

Station 3A function f is linear if its equation can be written in the form f is linear if its equation can be written in the form f f(f(f x(x( ) = mx + b, where m and b are real numbers. Use this information and the problem scenario that follows to answer the questions. You may use a calculator.

The cost of a sweatshirt is linearly related to the number of sweatshirts ordered. If you buy 100 sweatshirts, then the cost per sweatshirt is $19. However, if you buy 250 sweatshirts, then the cost per sweatshirt is only $17.

1. You are given information that determines two points in the function. If x represents the number of sweatshirts and y represents the cost per sweatshirt, write the two ordered pairs represented in the problem scenario above.

2. What is the slope of the function?

3. Find a function which relates the number of sweatshirts and the cost per sweatshirt. Show your work.

4. What would the cost per sweatshirt be for 500 sweatshirts? Explain.

5. What would the cost per sweatshirt be for 60 sweatshirts? Explain.




Station 4You will be given a number cube. As a group, roll the number cube and write the result in the first box. Repeat this process until all the boxes contain a number.

{( , ), ( , ), ( , ), ( , )}

1. What is the domain of this relation?

2. What is the range of this relation?

3. Is this relation a function? Why or why not?

For problems 4–6, state the domain, range, and whether the relation is a function. Include your reasoning.

4. {(2, 5), (3, 10), (–1, 2), (4, 5)}

5. {(10, 7), (3, 7), (10, 5), (7, 2)}

6. {(–14, 8), (17, 8), (14, –9), (15, 17)}

Formulas

© Walch EducationF-1

Formulas

Symbols

≈ Approximately equal to

≠ Is not equal to

a Absolute value of a

a Square root of a

General

(x, y) Ordered pair

(x, 0) x-intercept

(0, y) y-intercept

Linear Equations

my y

x x=

−−

2 1

2 1

Slope

ax + b = c One variable

y = mx + b Slope-intercept form

ax + by = c General form

y – y1 = m(x – x

1) Point-slope form

ALGEBRA

Exponential Equations

y = abx General form

y abx

t= Exponential equation

y = a(1 + r)t Exponential growth

y = a(1 – r)t Exponential decay

A Pr

n

nt

= +

1 Compounded interest formula

Compounded… n (number of times per year)

Yearly/annually 1

Semiannually 2

Quarterly 4

Monthly 12

Weekly 52

Daily 365

Arithmetic Sequences

an = a

1 + (n – 1)d Explicit formula

an = a

n –1 + d Recursive formula

Geometric Sequences

an = a

1 • rn – 1 Explicit formula

an = a

n – 1 • r Recursive formula

Functions

f(x) Notation, “f of x”

f(x) = mx + b Linear function

f(x) = bx + k Exponential function

(f + g)(x) = f(x) + g(x) Addition

(f – g)(x) = f(x) – g(x) Subtraction

(f • g)(x) = f(x) • g(x) Multiplication

(f ÷ g)(x) = f(x) ÷ g(x) Division

Formulas

F-2© Walch EducationFormulas

Properties of Equality

Property In symbols

Reflexive property of equality a = a

Symmetric property of equality If a = b, then b = a.

Transitive property of equality If a = b and b = c, then a = c.

Addition property of equality If a = b, then a + c = b + c.

Subtraction property of equality If a = b, then a – c = b – c.

Multiplication property of equality If a = b and c ≠ 0, then a • c = b • c.

Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c.

Substitution property of equality If a = b, then b may be substituted for a in any expression containing a.

Properties of Operations

Property General rule

Commutative property of addition a + b = b + a

Associative property of addition (a + b) + c = a + (b + c)

Commutative property of multiplication a • b = b • a

Associative property of multiplication (a • b) • c = a • (b • c)

Distributive property of multiplication over addition a • (b + c) = a • b + a • c

Properties of Inequality

Property

If a > b and b > c, then a > c.

If a > b, then b < a.

If a > b, then –a < –b.

If a > b, then a ± c > b ± c.

If a > b and c > 0, then a • c > b • c.

If a > b and c < 0, then a • c < b • c.

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c.

Laws of Exponents

Law General rule

Multiplication of exponents

bm • bn = bm + n

Power of exponentsb bm n mn( ) =

bc b cn n n( ) =

Division of exponentsb

bb

m

nm n= −

Exponents of zero b0 = 1

Negative exponents bb

nn=− 1

and b

bnn=−

1

Formulas

© Walch EducationF-3

Formulas

DATA ANALYSIS

IQR = Q 3 – Q

1Interquartile range

Q 1 – 1.5(IQR) Lower outlier formula

Q 3 + 1.5(IQR) Upper outlier formula

y – y0

Residual formula

GEOMETRY

Symbols

d ABC( ) Arc length

∠ Angle

Circle

≅ Congruent

PQ� ��

Line

PQ Line Segment

PQ� ��

Ray

Parallel

⊥ Perpendicular

• Point

Triangle

A′ Prime

° Degrees

Translations

T(h, k)

= (x + h, y + k) Translation

Reflections

rx-axis

(x, y) = (x, –y) Through the x-axis

ry-axis

(x, y) = (–x, y) Through the y-axis

ry = x

(x, y) = (y, x) Through the line y = x

Rotations

R90

(x, y) = (–y, x) Counterclockwise 90° about the origin

R180

(x, y) = (–x, –y) Counterclockwise 180° about the origin

R270

(x, y) = (y, –x) Counterclockwise 270° about the origin

Congruent Triangle Statements

Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA)

B

A

C X

Z Y

FD

E

V

TW

J

G

H

S

Q

R

≅ABC XYZ ≅DEF TVW ≅GHJ QRS

Formulas

F-4© Walch EducationFormulas

Pythagorean Theorem

a2 + b2 = c2

Area

A = lw Rectangle

A bh=1

2 Triangle

Distance Formula

d x x y y= − + −( ) ( )2 12

2 12

Distance formula

MEASUREMENTS

Length

Metric

1 kilometer (km) = 1000 meters (m)

1 meter (m) = 100 centimeters (cm)

1 centimeter (cm) = 10 millimeters (mm)Customary

1 mile (mi) = 1760 yards (yd)

1 mile (mi) = 5280 feet (ft)

1 yard (yd) = 3 feet (ft)

1 foot (ft) = 12 inches (in)

Volume and Capacity

Metric

1 liter (L) = 1000 milliliters (mL)Customary

1 gallon (gal) = 4 quarts (qt)

1 quart (qt) = 2 pints (pt)

1 pint (pt) = 2 cups (c)

1 cup (c) = 8 fluid ounces (fl oz)

Weight and Mass

Metric

1 kilogram (kg) = 1000 grams (g)

1 gram (g) = 1000 milligrams (mg)

1 metric ton (MT) = 1000 kilograms (kg)

Customary

1 ton (T) = 2000 pounds (lb)

1 pound (lb) = 16 ounces (oz)

Glossary

English Unit/Lesson EspañolA

arithmetic sequence a linear function with a domain of positive consecutive integers in which the difference between any two consecutive terms is equal

2.14 secuencia aritmética función lineal con dominio de enteros consecutivos positivos, en la que la diferencia entre dos términos consecutivos es equivalente

Bbase the factor being multiplied together

in an exponential expression; in the expression ab, a is the base

2.2 base factor que se multiplica en forma conjunta en una expresión exponencial; en la expresión ab, a es la base

boundary line the graph of the line that represents a linear inequality and that divides the coordinate plane into two half planes, one of which contains all the solutions of the inequality

2.9 línea de límite la gráfica de la línea que representa una desigualdad lineal y que divide el plano de coordenadas en dos medios planos, uno de los cuales contiene todas las soluciones de la desigualdad

Ccommon difference the number added to

each consecutive term in an arithmetic sequence

2.14 diferencia común número sumado a cada término consecutivo en una secuencia aritmética

constraint a restriction or limitation on any of the variables in an equation or inequality

2.9 limitación una restricción o limitación de cualquiera de las variables en una ecuación o desigualdad

coordinate plane a plane determined by a set of two number lines, called the axes, that intersect at right angles

2.8 plano de coordenadas un plano determinado por un conjunto de dos líneas numéricas, llamadas los ejes, que se cruzan en ángulos rectos

curve the graphical representation of the solution set for y = f(f(f x(x( ); in the special case of a linear equation, the curve will be a line

2.7 curva representación gráfica del conjunto de soluciones para y = f(f(f x(x( ); en el caso especial de una ecuación lineal, la curva será una recta

Ddependent variable generally labeled

on the y-axis; the quantity that is based on the input values of the independent variable

2.8 variable dependiente generalmente designada en el eje y; cantidad que se basa en los valores de entrada de la variable independiente

© Walch Education© Walch EducationG-1

Glossary

Glossary

English Unit/Lesson EspañolE

equation a mathematical sentence that uses an equal sign (=) to show that two quantities are equal

2.13 ecuación declaración matemática que utiliza el signo igual (=) para demostrar que dos cantidades son equivalentes

explicit function a function in which the dependent variable can be written in terms of the independent variable; f(f(f x(x( ) = 2x is an explicit function, where x is the independent variable and f(f(f x(x( ) is the dependent variable

2.13 función explícita una función en la que la variable dependiente se puede escribir en términos de la variable independiente; f(f(f x(x( ) = 2x es una función explícita, donde x es la variable independiente y f(f(f x(x( ) es la variable dependiente

expression a combination of variables, quantities, and mathematical operations; 4, 8x, and b + 102 are all expressions

2.13 expresión combinación de variables, cantidades y operaciones matemáticas; 4, 8x y b + 102 son todas expresiones

Ffunction a relation in which each element

in the domain is mapped onto exactly one element in the range; that is, for every value of x, there is exactly one value of y

2.3 función relación en la que cada elemento de un dominio se combina con exactamente un elemento del rango; es decir, para cada valor de x, existe exactamente un valor de y

Hhalf plane a planar region containing all

points that lie on one side of a boundary line; one-half of a plane

2.9 semiplano una región plana que contiene todos los puntos que se encuentran en un lado de una línea de límite; la mitad de un avión

Iinclusive when the points in a plane

that lie along the boundary line of an inequality are included in the solution

2.9 inclusivo cuando los puntos en un plano que se encuentran a lo largo de la línea de límite de una desigualdad se incluyen en la solución

independent variable generally labeled on the x-axis; the quantity that changes based on values chosen

2.8 variable independiente generalmente designada en el eje x; cantidad que cambia según valores seleccionados

intercept the value of the x- or y-coordinate where a line or curve intersects the x- or y-axis, respectively

2.9 intersección valor de la coordenada x o ydonde una línea o curva interseca el eje xo y, respectivamente

G-2© Walch Education© Walch EducationGlossary

Glossary

English Unit/Lesson EspañolL

linear equation a first-degree equation that can be written in the form ax + by = c, where a, b, and c are c are c rational numbers; when written as y = mx + b, m is the slope of the line, and b is its y-intercept. The graph of a linear equation is a straight line.

2.72.8

ecuación lineal ecuación de primer grado que puede expresarse en la forma ax + by = c, donde a, b y c son números c son números cracionales; cuando se expresa como y = mx + b, m es la pendiente de la recta y b es el intercepto de y. La representación gráfica de una ecuación lineal es una línea recta.

linear function a first-degree equation that can be written in the form f(f(f x(x( ) = mx + b, in which m is the slope of the line and b is the y-intercept. The graph of a linear function is a straight line.

2.32.11

función lineal una ecuación de primer grado que puede expresarse en la forma f(f(f x(x( ) = mx + b, en la que m es la pendiente de la recta y b es el intercepto de y. El gráfico de una función lineal es una línea recta.

N

non-inclusive when the points in a plane that lie along the boundary line of an inequality are not included in the solution

2.9 no inclusivo cuando los puntos en un plano que se encuentran a lo largo de la línea de límite de una desigualdad no están incluidos en la solución

O

ordered pair the coordinates of a point in a coordinate plane, (xa coordinate plane, (xa coordinate plane, ( , y) where the order is significant

2.7 par ordenado coordenadas de un punto en un plano de coordenadas, (x (x ( , y), en los que el orden es significativo

P

parameter a constant in a function that determines the specific graph of the function but not the type of the function

2.6 parámetro una constante en una función que determina el gráfico específico de la función pero no el tipo de la función

point-slope form the form y – y1 = m(x(x( – x1), where m is the slope, and (xand (xand ( 1, y1) is a point on the line

2.3 forma punto-pendiente la forma y – y1 = m(x(x( – x1), donde m es la pendiente y (xy (xy ( 1, y1) es un punto de la recta

proportional having a constant ratio to another quantity

2.4 proporcional que tiene una proporción constante con otra cantidad

© Walch Education© Walch EducationG-3

Glossary

Glossary

English Unit/Lesson EspañolS

slope the measure of the rate of change

of one variable with respect to another

variable; slope = m = rise

run2 1

2 1

�

�

−−

= =y y

x x

y

x

rise

run2 1

2 1

�

�

−−

= =y y

x x

y

x;

the slope in the equation y = y = y mx + mx + mx b is m

2.32.42.8

pendiente medida de la tasa de cambio

de una variable con respecto a otra;

pendiente = m = rise

run2 1

2 1

�

�

−−

= =y y

x x

y

x

rise

run2 1

2 1

�

�

−−

= =y y

x x

y

x;

la pendiente en la ecuación y = mx + mx + mx b es mslope-intercept form of a linear

equation the form y = mx + b, where m is the slope of the line and b is the y-intercept

2.3 forma pendiente-intersección de una ecuación lineal la forma y = mx + b, donde m es la pendiente y b es el punto de intersección con el eje y

solution set the value or values that make a sentence or statement true; the set of ordered pairs that represent all of the solutions to an equation or a system of equations

2.7 conjunto de soluciones valor o valores que hacen verdadera una afirmación o declaración; conjunto de pares ordenados que representa todas las soluciones para una ecuación o sistema de ecuaciones

Uunit rate a ratio of two measurements, the

second of which is 12.4 tasa unitaria una proporción de dos

medidas, de las que la segunda es 1

Vvariable a letter used to represent an

unknown value or a value that changes2.13 variable una letra utilizada para

representar un valor desconocido o un valor que cambia

Xx-interceptx-interceptx the x-coordinate of the point

where a line or a curve intersects the x-axis2.82.9

intersección x la coordenada x del punto en que una recta o curva corta el eje x

Yy-intercepty-intercepty the y-coordinate of the point

where a line or a curve intersects the y-axis2.32.8

intersección y la coordenada y del punto en que una recta o curva corta el eje y

G-4© Walch Education© Walch EducationGlossary

North Carolina Math 1

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